On the Geometry of Principal Homogeneous Spaces
aa r X i v : . [ m a t h . AG ] O c t On the Geometry of Principal Homogeneous Spaces
A. J. de Jong ∗ and Robert FriedmanNovember 3, 2018 Introduction
Let k be an algebraically closed field, let π : X → B be an elliptic surfacedefined over k , and let X K be the generic fiber of π , which is an ellipticcurve defined over the field K = k ( B ), the function field of B . If f : Y → B is a genus one fibration locally isomorphic to X (in the ´etale topology on B ),then Y corresponds to a principal homogeneous space Y K over X K which iseverywhere locally trivial. The goal of this paper is to study the geometryof such surfaces. Of course, the main interest is when X , B and Y areinstead defined over a finite field F . In this case, the set of isomorphismclasses of everywhere locally trivial principal homogeneous spaces for X K isconjectured to be finite. This is known to hold in case X ∼ = D × B , where D is an elliptic curve D defined over F , or X is a rational surface, by workof Milne [18, 19], or X is a K F , by work of Artin andSwinnerton-Dyer [2]. Since the appearance of [2], little progress has beenmade in the function field case, and it is our hope that the geometric studyof principal homogeneous spaces, over an algebraically closed field, may givesome clues as to how to attack the finiteness problem over finite fields.Suppose that f : Y → B is a genus one fibration, everywhere locallytrivial. Let n be the smallest positive integer such that there exists an n -section of f , in other words an irreducible curve D ⊆ Y whose intersectionnumber with a fiber of f is n . Fix a divisor D on Y , not necessarily effec-tive, such that the degree of the restriction O Y ( D ) to a smooth fiber is n .For simplicity, we also assume in the introduction that every fiber of f , orequivalently π , is irreducible. In this case, D is specified by its restrictionto a generic fiber up to adding an integral combination of smooth fibers.In particular, the rank n vector bundle f ∗ O Y ( D ) on the base curve B is ∗ The first author was supported in part by NSF grant B . Of course, by rela-tive duality, it is essentially equivalent to consider the rank n vector bundle R f ∗ O Y ( − D ) ∼ = ( f ∗ O Y ( D )) ∨ ⊗ ω − . Here ω is an invertible sheaf on B pulling back to the relative dualizing sheaves ω Y/ P , resp. ω X/ P on Y , resp. X . If B ∼ = P , then we can write R f ∗ O Y ( − D ) = n M i =1 O P ( α i )for unique integers α ≤ α ≤ · · · ≤ α n . Geometrically, the significanceof the bundle R f ∗ O Y ( − D ) is as follows: for n ≥
3, the divisor D is rel-atively very ample and induces an embedding Y in the projective bundle P ( R f ∗ O Y ( − D )) over B (and realizes Y as a double cover of P ( R f ∗ O Y ( − D ))in case n = 2).Ideally the vector bundle R f ∗ O Y ( − D ) should be fairly well behaved. Inparticular it is natural to ask if it is semistable. This is not possible for manyreasons. For example, if B ∼ = P , then a bundle L ni =1 O P ( α i ) is semistableif and only if α = · · · = α n . But D = − P i α i + ( n − d , where d = χ ( Y ; O Y ), and D mod 2 n , which is clearly an invariant of the restriction of D to the generic fiber, is an essentially topological invariant. In particularone can show that many values of D mod 2 n occur, so that R f ∗ O Y ( − D )cannot in general be semistable. However, again in the case B ∼ = P , onecould ask if, in the above notation, | α i − α j | ≤ i, j . Equivalently, upto a twist, is R f ∗ O Y ( − D ) always of the form O k P ⊕ O P ( − n − k ? Definethe pair ( Y, D ) to be rigid if B = P and there exists an integer t such that f ∗ O Y ( D ) ∼ = O P ( t ) k ⊕ O P ( t − n − k . Thus (for the case B ∼ = P ), if ( Y, D )is a rigid pair, then R f ∗ O Y ( − D ) is optimal in various senses: as a bundleover P , it is both rigid and generic in moduli.It is easy to see that not every pair ( Y, D ) is rigid. However, in somesense, every pair is not too far from being rigid:
Theorem 0.1.
Let f : Y → B ∼ = P , D , and n be as above. Let d = χ ( X ; O X ) , and suppose that the characteristic of k does not divide n . Let R f ∗ O Y ( − D ) = L ni =1 O P ( α i ) with α ≤ α ≤ · · · ≤ α n . Then ≤ α n − α ≤ d/ . In fact, we prove a slightly better bound. In the cases where one cancompute all possible examples by hand, namely n = 2 , α i , and to generalize to arbitrary base curves, recall that, for a rank n bundle2n a curve B , the slope µ ( V ) of V is the rational number deg( V ) /n , and thebundle V is semistable if, for all subbundles W of V with 0 < rank( W ) < n , µ ( V ) ≥ µ ( W ). We then prove the following theorem: Theorem 0.2.
Let f : Y → B , D , n , and d be as above. Suppose that thecharacteristic of k does not divide n . Let W be a subbundle of R f ∗ O Y ( − D ) of rank r , < r < n , and let e = gcd( r + 1 , n ) . Then µ ( R f ∗ O Y ( − D )) − µ ( W ) ≥ − (cid:18) r ( n − r ) + ( e − nr (cid:19) d. Essentially trivial manipulations give a corresponding result for quotientbundles:
Corollary 0.3.
With the above notation and hypotheses, suppose that Q is a quotient bundle of R f ∗ O Y ( − D ) of rank r , < r < n , and let e =gcd( n − r + 1 , n ) . Then µ ( R f ∗ O Y ( − D )) − µ ( Q ) ≤ (cid:18) r ( n − r ) + ( e − nr (cid:19) d. Remark 0.4. (i) There are slightly sharper bounds in case B ∼ = P . Usingthese, and considering the appropriate rank one subbundle and quotientbundle of R f ∗ O Y ( − D ), one can check that Theorem 0.2 and Corollary 0.3essentially imply the bounds of Theorem 0.1.(ii) One can drop the hypothesis that the characteristic of k does not divide n , but the bounds are not as strong in this case.Theorem 0.2 or the equivalent Corollary 0.3 say that, while the rank n bundle R f ∗ O Y ( − D ) may not be semistable, its failure to be semistable canbe controlled in a fairly precise way.Very roughly speaking, the idea behind the proofs of Theorem 0.1 andTheorem 0.2 is as follows. Let V be a vector bundle of rank r on a smoothsurface S . Define ∆( V ) = 2 rc ( V ) − ( r − c ( V ) . Then in characteristiczero, Bogomolov’s inequality says that, if V is semistable, then ∆( V ) ≥ V has rank 2). The link between the case where V is semistable and the casewhere the restriction of V to the generic fiber is stable is as follows: For a3enus one fibration f : Y → B and a vector bundle V on Y , the semistabil-ity of V with respect to an ample divisor numerically equivalent to one ofthe form H + tF , where F is the numerically equivalence class of a generalfiber and t ≫
0, is closely related to the semistability of the restriction of V to the (geometric) generic fiber of f , which is an elliptic curve defined overthe algebraic closure of the function field of B [10, 11]. For example, let K = Spec k ( B ), where k ( B ) is the function field of B , and suppose that therestriction V K of V to the generic fiber Y K of f , which is a curve of genusone defined over the non-algebraically closed field K , is stable. Then it iseasy to see that V is semistable with respect to some ample divisor of theform given above.We then prove the following inequality: Theorem 0.5.
Let f : Y → B be a genus one fibration over the algebraicallyclosed field k with χ ( Y ; O Y ) = d , and let V be a vector bundle of rank r whose restriction to the generic fiber of f is stable. Let n be the degree ofthe restriction of V to the generic fiber of f , and let e = gcd( n, r ) . Supposethat the characteristic of k does not divide r . Then ∆( V ) ≥ ( r − e ) d. (There is also a weaker inequality than that of Theorem 0.5 which holdswith no assumption on the characteristic of k .)Given Theorem 0.5, the proof of Theorem 0.1 and Theorem 0.2 becomesa matter of constructing the appropriate vector bundles V . There are manyapproaches to doing so. For example, Theorem 0.2 follows by consideringcertain “universal extensions.” Other methods for constructing bundles leadto results on the vanishing of H ( Y ; O Y ( D )) or base point free and embed-ding theorems for the linear system | D | on Y , following methods of Mumford[20] and Reider [22].While Theorem 0.1 and Theorem 0.2 are valid for all principal homoge-neous spaces, one can ask the following question: Fixing a generic ellipticsurface f : X → B , what can one say about the vector bundles R f ∗ O Y ( − D )for all pairs ( Y, D ), where D is an n -section. In the case B ∼ = P , and k = C ,a degeneration argument shows: Theorem 0.6.
Let X be a generic elliptic surface over P . Then every pair ( Y, D ) such that X is the Jacobian surface of Y is rigid. It is very likely that the methods of proof can be extended to handle thecase of positive characteristic as well.4he contents of this paper are as follows: In Section 1, we establish theBogomolov type inequality of Theorem 0.5. In Section 2, we construct vec-tor bundles for which we can apply this inequality and prove Theorem 0.2.In Section 3, we construct examples for small values of n and show that ourbounds are sharp. Section 4 studies the group of components of the coarsemoduli space of genus one fibrations over P . Although we have statedthe result in terms of coarse moduli spaces, it could also be formulated viastacks. Unfortunately, in the determination of the monodromy, we eventu-ally have to assume that k = C . However, we expect that similar resultshold in the case of positive characteristic as well. Somewhat inconsistentlywe have devoted a fair amount of space to a proof in all characteristics ofa (generalization of) a result of Artin and Swinnerton-Dyer; the proof isstraightforward over C . Finally, in Section 5, we prove Theorem 0.6 via adegeneration argument, induction, and a result concerning rational ellipticsurfaces. Conventions:
All schemes are separated and of finite type over k , an al-gebraically closed field. If X is a scheme and F is a sheaf on X , we shalluse H i ( X ; F ) to denote the usual sheaf cohomology in case F is coherent,´etale cohomology in case F is a sheaf of the form µ n or one of its variants,or the Betti cohomology of X ( C ) in case X is defined over C and F = Z .The meaning should be clear from the context.A genus one fibration f : Y → B without multiple fibers , or more brieflya genus one fibration , is a smooth projective surface Y , together with amorphism f to a smooth projective curve B , such that no exceptional curveof Y is contained in a fiber of f ( Y is relatively minimal), every fiber of f has trivial dualizing sheaf (hence is of arithmetic genus one), and f has nomultiple fibers (if U denotes the Zariski open subset of Y where f is smooth,then f ( U ) = B ).Let f : Y → B be a genus one fibration. The order of Y is the smallestpositive integer n such that there exists an element D in Pic Y with D · F = n ,where F is the class of a general fiber of f .A elliptic surface π : X → B is a genus one fibration endowed with asection σ : B → X . Definition 1.1.
Let V be a vector bundle of rank r on a smooth projectivesurface S . Define ∆( V ) = 2 rc ( V ) − ( r − c ( V ) . By the splitting principle,5( V ) = c ( End V ).According to Bogomolov’s inequality, if k has characteristic zero and V is semistable, then ∆( V ) ≥
0. To deal with the analogue of this inequality inpositive characteristic, recall that, if E K is a curve defined over a (not neces-sarily algebraically closed) field K , and V K is a locally free sheaf on E K , then V K is stable (resp. semistable )if, for every subsheaf W K of V K (hence definedover K ), we have µ ( W K ) < µ ( V K ) (resp. µ ( W K ) ≤ µ ( V K )), where by defi-nition the slope µ ( W K ) of a locally free sheaf on E K is deg( W K ) / rank W K .Standard arguments show:1. If V K is stable and ϕ ∈ End V K , then either ϕ = 0 or ϕ is an isomor-phism, and hence that End V K is a division algebra of finite dimensionover K .2. If V K is stable and ¯ K is the algebraic closure of K , then the pullback V ¯ K of V K to E K × Spec K ¯ K is semistable.The following theorem is then the first main result of this paper: Theorem 1.2.
Let f : Y → B be a genus one fibration with χ ( Y ; O Y ) = d ,and let V be a vector bundle of rank r whose restriction to the generic fiberof f is stable. Let n be the degree of the restriction of V to the generic fiberof f , and let e = gcd( n, r ) . Suppose that the characteristic of k does notdivide e . Then ∆( V ) ≥ ( r − e ) d. If moreover B ∼ = P , then ∆( V ) ≥ ( r − e ) d + 2( e − . Proof.
We begin with the following:
Proposition 1.3.
With notation as above, the sheaf f ∗ End V is a locallyfree sheaf of commutative O B -algebras of rank e .Proof. Since f ∗ End V is a torsion free O B -module, it is locally free. Let K = k ( B ) be the function field of B , and let Y K = Y × B Spec K . Then Y K is a genus one curve over Spec K . Let V K be the corresponding vectorbundle. It suffices to prove that the restriction of f ∗ End V to Spec K isa commutative K -algebra of dimension e . Clearly, this restriction is justEnd V K . By hypothesis, V K is stable and hence D = End V K is a divisionalgebra of finite dimension over K . Let F be the center of D . Then F is a6nite extension of K and thus is a finitely generated field of transcendencedegree one over the algebraically closed field k . By Tsen’s theorem, theBrauer group of F is trivial. Thus the element of the Brauer group of F defined by D is trivial, so that D = F . In particular, D is commutative.Let ¯ K be the algebraic closure of K . To see that the rank of D is e , itsuffices to prove the corresponding statement for End V ¯ K , where V ¯ K is thepullback of V K to the elliptic curve Y ¯ K = Y K × Spec K Spec ¯ K . The bundle V ¯ K is semistable, and we have the following well-known lemma: Lemma 1.4.
Let E be an elliptic curve defined over an algebraically closedfield ¯ K , and let V ¯ K be a semistable vector bundle on E of rank r and degree n . Let e = gcd( n, r ) . Then dim ¯ K End V ¯ K ≥ e , and dim ¯ K End V ¯ K = e ifand only if End V ¯ K is commutative.Proof. Let n = n/e and r = r/e . We recall some consequences of Atiyah’sclassification of vector bundles V ¯ K over an elliptic curve E = E ¯ K definedover an algebraically closed field ([3], Theorem 7 and its corollary): Supposethat n and r are two relatively prime positive integers and that λ is a linebundle on E of degree n . Then1. There is a unique stable bundle V n ,r , λ on E of rank r and degree n such that det V n ,r , λ = λ .2. For every positive integer d , there is a unique indecomposable vec-tor bundle V n ,r ,d ; λ on E of rank dr and degree dn , all of whosesuccessive Jordan-H¨older quotients are isomorphic to V n ,r , λ .3. Hom( V n ,r ,d ; λ , V n ,r ,d ; λ ) ∼ = ( , if λ = λ ;¯ K [ t ] / ( t k ) , k = min( d , d ) , if λ = λ .
4. If V ¯ K is a semistable vector bundle of rank r and degree n , with e =gcd( n, r ) and n/e = n , r/e = r , then V ¯ K ∼ = M λ V ( λ ) , where V ( λ ) is the uniquely defined summand of V ¯ K which is the (notnecessarily direct) sum of all indecomposable summands of V ¯ K of theform V n ,r ,d ; λ and hence V ( λ ) ∼ = L i V n ,r ,d i ; λ .7ith V ¯ K ∼ = L λ V ( λ ) as above, clearlyEnd ¯ K V ¯ K ∼ = M λ End ¯ K V ( λ ) . If V ( λ ) ∼ = L i V n ,r ,d i ; λ , with rank V ( λ ) = P i d i r = d λ r , then r = er =( P λ d λ ) r , and hence e = P λ d λ . Direct computation then shows thatEnd ¯ K V ( λ ) is commutative if and only if V ( λ ) ∼ = V n ,r ,d ; λ is indecompos-able, and also that dim End ¯ K V ( λ ) ≥ d λ = P i d i , with equality holding ifand only if V ( λ ) ∼ = V n ,r ,d ; λ is indecomposable. Combining the two state-ments gives the proof of the lemma.To complete the proof of the proposition, since End V ¯ K = End V K ⊗ K ¯ K is commutative, it follows from Lemma 1.4 that dim ¯ K End V ¯ K = e , andhence dim K End V K = e as well. Thus f ∗ End V is a locally free sheaf ofcommutative O B -algebras of rank e .Let C = Spec f ∗ End V be the scheme over B associated to the coher-ent sheaf of commutative O B -algebras f ∗ End V . Thus there is a finite flatmorphism g : C → B of degree e . The scheme C is sometimes called the spectral cover of B associated to the bundle V . Since the restriction of f ∗ End V to Spec K is a field, C is an integral scheme of dimension one, and g ∗ O C = f ∗ End V . Let γ = deg f ∗ End V = deg g ∗ O C . Lemma 1.5.
Suppose that B is a smooth projective curve and that g : C → B is a finite flat separable morphism of degree e , for example suppose thatthe characteristic of k does not divide e . Set γ = deg g ∗ O C . Then γ ≤ . Ifin addition B ∼ = P , then γ ≤ − ( e − .Proof. Let ν : e C → C be the normalization. By applying g ∗ to the exactsequence 0 → O C → ν ∗ O e C → Q → , where Q is some torsion sheaf, we see that deg g ∗ O C ≤ deg( g ◦ ν ) ∗ O e C , andhence it suffices to prove the result when C = e C is smooth. In this case, awell known formula says that 2 deg g ∗ O C = − b , where b ≥ B . Thus γ = − b/ ≤
0. If inaddition g ( B ) = 0, then the Riemann-Hurwitz formula says that 2 g ( C ) − e (2 g ( B ) − b = − e + b . Hence γ = − b/ − ( e + g ( C ) − ≤ − ( e − χ ( Y ; End V ) intwo different ways. Since
End V is a vector bundle of rank r on Y with8 ( End V ) = 0 and c ( End V ) = ∆( V ), the Riemann-Roch theorem forvector bundles and the fact that χ ( Y ; O Y ) = d imply that χ ( Y ; End V ) = − ∆( V ) + r d. On the other hand, by the Leray spectral sequence, χ ( Y ; End V ) = χ ( B ; f ∗ End V ) − χ ( B ; R f ∗ End V ) . Applying Riemann-Roch to the rank e vector bundle f ∗ End V = g ∗ O C on B gives χ ( B ; f ∗ End V ) = γ + e (1 − g ), where g = g ( B ). Applying relativeduality to the morphism f : Y → B , which is a local complete intersectionmorphism of relative dimension one, gives( R f ∗ End V ) ∨ ∼ = f ∗ (( End V ) ∨ ⊗ ω Y/B ) . Since f : Y → B is a genus one fibration with no multiple fibers, ω Y/B = f ∗ ω ,where ω is a line bundle on B of degree d , and ( End V ) ∨ ∼ = End V . Thus:( R f ∗ End V ) ∨ ∼ = ( f ∗ End V ) ⊗ ω. This says that ( R f ∗ End V ) ∨∨ ∼ = [( f ∗ End V ) ∨ ⊗ ω − ], and hence that R f ∗ End V ∼ = T ⊕ [( f ∗ End V ) ∨ ⊗ ω − ] , where T is a torsion line bundle on B . Now f ∗ ( End V ) is a vector bundleon V of rank e and degree γ . Thus ( f ∗ End V ) ∨ is a vector bundle on V ofrank e and degree − γ , and ( f ∗ End V ) ⊗ ω − has rank e and degree − γ − ed .Hence χ ( B ; R f ∗ End V ) = ℓ ( T ) + ( − γ − ed ) + e (1 − g ) . Thus, if t = ℓ ( T ), χ ( Y ; End V ) = χ ( B ; f ∗ End V ) − χ ( B ; R f ∗ End V )= γ + e (1 − g ) − ( t + ( − γ − ed ) + e (1 − g ))= 2 γ − t + ed ≤ γ + ed. Thus − ∆( V ) + r d ≤ γ + ed , and after rearranging this gives∆( V ) ≥ r d − ed − γ ≥ ( r − e ) d. Moreover, if B ∼ = P , then∆( V ) ≥ r d − ed − γ ≥ ( r − e ) d + 2( e − . This concludes the proof of Theorem 1.2.9 emark 1.6.
Let g : C → B be the spectral cover of B corresponding to V . Clearly V is a module over f ∗ f ∗ End V = f ∗ g ∗ O C . If h : Y × B C → C and ν : Y × B C → Y are the natural morphisms, then by flat base change f ∗ g ∗ O C = ν ∗ h ∗ O C = ν ∗ O Y × B C . Thus V corresponds to a sheaf L over Y × B C , and it is easy to check that L is a torsion free sheaf on Y × B C ofrank r/e with ν ∗ L = V . As before, let f : Y → B be a relatively minimal elliptic fibration withgeneric fiber F , let n be the order of Y , i.e. the smallest positive integersuch that there exists a divisor on Y whose intersection number with F is n , and let D be a divisor on Y such that D · F = n . In case not all fibers of f are irreducible, we shall make the following assumption on the divisor D : Assumption: If C is a component of a fiber of f , then D · C ≥
0, i.e. D is f -nef.For example, this assumption is always satisfied if D is effective andirreducible. If ℓ is a divisor on B of degree t ≫
0, then D + f ∗ ℓ is effectiveby Riemann-Roch, and thus linearly equivalent to D ′ + P i C i , where D ′ is effective and irreducible by our assumptions on n . Hence, for arbitrary D , there always exists a D ′ whose restriction to the generic fiber is linearlyequivalent to D and which is f -nef.A related property of D is the following: Definition 2.1.
The divisor D with D · F = n is minimal if D is effectiveand, for every component C of a fiber of f , D − C is not effective.Clearly, if D is minimal, then every curve in | D | is reduced irreducible,and hence D is f -nef. Given an effective divisor D with D · F = n , it is easy tosee that there exists a minimal divisor D such that D is linearly equivalentto D + P i n i C i , where the C i are components of fibers and n i ≥
0: Let H be a fixed ample divisor. If D is not minimal, then there exists a component C of a fiber such that D = D − C is effective. If D is minimal, weare done, otherwise we can repeat this process. As 0 < H · D < H · D ,we cannot continue indefinitely, so we eventually produce a minimal divisor D n = D = D − P i n i C i as desired.An elementary calculation (for example, using Ramanujam’s lemma asin the proof of Proposition 4.6) shows that, if F is a fiber of f and λ isa line bundle on F of degree − n such that, for every component C of F ,deg( λ | C ) ≤
0, then H ( F ; λ ) = 0 and hence dim H ( F ; λ ) = n . Thus:10 emma 2.2. If D is f -nef, then R f ∗ O Y ( − D ) is locally free of rank n . Under our assumption that D is f -nef, an easy application of relative du-ality shows that the rank n bundle f ∗ O Y ( D ) on B is related to R f ∗ O Y ( − D )as follows: R f ∗ O Y ( − D ) ∼ = [ f ∗ O Y ( D )] ∨ ⊗ ω − , where as before ω is the line bundle on B such that f ∗ ω = ω Y/B , and hencedeg ω = χ ( Y ; O Y ) = d . Lemma 2.3.
Let δ = deg R f ∗ O Y ( − D ) . Then D = − δ − ( n + 2) d. Thus, the slope µ = µ ( R f ∗ O Y ( − D )) of R f ∗ O Y ( − D ) is µ = δn = − D n − ( n + 2) d n . Proof.
By Riemann-Roch, χ ( Y ; O Y ( − D )) = ( D + D · K Y ) + χ ( Y ; O Y ) = ( D + n ( d + 2 g − d . On the other hand, χ ( Y ; O Y ( − D )) = χ ( B ; R f ∗ O Y ( − D )) − χ ( B ; R f ∗ O Y ( − D )) = − ( δ + n (1 − g )) , using Riemann-Roch on B for the vector bundle R f ∗ O Y ( − D ). Thus D + ( n + 2) d = − δ, proving the first statement of the lemma, and the second is then clear. Let W be a subbundle of R f ∗ O Y ( − D ) of rank r and degree δ W . Thus W has slope µ ( W ) = δ W /r . We calculate the Ext group Ext ( f ∗ W, O Y ( − D )):Ext ( f ∗ W, O Y ( − D )) = H ( Y ; f ∗ W ∨ ⊗ O Y ( − D ))= H ( B ; W ∨ ⊗ R f ∗ O Y ( − D )) = Hom( W, R f ∗ O Y ( − D )) . Thus, the inclusion W → R f ∗ O Y ( − D ) defines an extension V , givenby 0 → O Y ( − D ) → V → f ∗ W → , with the property that the induced coboundary homomorphism f ∗ f ∗ W = W → R f ∗ O Y ( − D ) is the given inclusion. One easily calculates the Chernclasses of V : 11 emma 2.4. With V as above, the rank of V is r + 1 and the degree of therestriction of V to the generic fiber of f is − n . Moreover, c ( V ) ≡ − D + δ W · F ; c ( V ) = − nδ W . Hence ∆( V ) = 2( r + 1)( − nδ W ) − r ( D − nδ W ) = − nδ W − rD = − nδ W + 2 rδ + r ( n + 2) d. Lemma 2.5.
The restriction V K of V to the generic fiber Y K of f is stable.Proof. There is an exact sequence0 → O Y K ( − D ) → V K → O rY K → , and the induced homomorphism H ( Y K ; O rY K ) → H ( Y K ; O Y K ( − D )) is in-jective. The slope of V K is − n/ ( r + 1). Let S be a subbundle of V K , with0 < rank S < r + 1. For the purposes of checking the stability of V K , we mayassume that S is semistable. If deg S ≥
0, then the induced homomorphism S → O rY K is not zero. Thus the image of S in O rY K has degree ≥
0, hence hasdegree 0 and is a summand of O rY K . Thus H ( Y K ; S ) is a nonzero summandof H ( Y K ; O rY K ) which maps to zero in H ( Y K ; O Y K ( − D )). This contradictsthe injectivity of the homomorphism H ( Y K ; O rY K ) → H ( Y K ; O Y K ( − D )).Thus, deg S <
0. It follows that deg S = − kn for some positive integer k ,and hence S has slope − kn/t , where 0 < t < r + 1. In this case, the slopeof S is clearly < − n/ ( r + 1). Thus V K is stable. Theorem 2.6.
With assumptions as above, suppose that the characteristicof k does not divide n . Let W be a subbundle of R f ∗ O Y ( − D ) of rank r , < r < n , and let e = gcd( r + 1 , n ) . Set µ = µ ( R f ∗ O Y ( − D )) . Then µ − µ ( W ) ≥ − (cid:18) r ( n − r ) + ( e − nr (cid:19) d. If moreover B ∼ = P , then µ − µ ( W ) ≥ − (cid:18) r ( n − r ) + ( e − nr (cid:19) d + ( e − nr . roof. Lemma 2.5 and the hypothesis that char k does not divide n implythat the assumptions of Theorem 1.2 hold. Thus ∆( V ) ≥ (( r + 1) − e ) d .By Lemma 2.4, we have − nδ W + 2 rδ + r ( n + 2) d ≥ (( r + 1) − e ) d. Rearranging gives2 rδ − nδ W ≥ ( r + 2 r + 1 − e − rn − r ) d = ( r ( r − n ) + (1 − e )) d. Dividing by 2 nr gives the inequality. The last statement follows from theconclusions of Theorem 1.2 in case B ∼ = P . Corollary 2.7.
With assumptions as above, suppose that the characteristicof k does not divide n . Let Q be a quotient bundle of R f ∗ O Y ( − D ) of rank r , < r < n , and let e = gcd( n − r + 1 , n ) . Set µ = µ ( R f ∗ O Y ( − D )) . Then µ − µ ( Q ) ≤ (cid:18) r ( n − r ) + ( e − nr (cid:19) d. If moreover B ∼ = P , then µ − µ ( Q ) ≤ (cid:18) r ( n − r ) + ( e − nr (cid:19) d − ( e − nr . Proof.
Let W be the kernel of the surjection R f ∗ O Y ( − D ) → Q , so that W has rank n − r . Then( n − r ) µ ( W ) + rµ ( Q ) = nµ = ( n − r ) µ + rµ. Thus µ ( Q ) − µ = (cid:18) n − rr (cid:19) ( µ − µ ( W )) . By Theorem 2.6, µ ( Q ) − µ ≥ − (cid:18) n − rr (cid:19) (cid:18) r ( n − r ) + ( e − n ( n − r ) (cid:19) d = − (cid:18) r ( n − r ) + ( e − nr (cid:19) d. Reversing the signs gives the first conclusion of the corollary, and the secondis similar. 13n case B ∼ = P , we can make the above inequalities more concrete asfollows: Corollary 2.8.
Suppose that B ∼ = P and that the characteristic of k doesnot divide n . Let R f ∗ O Y ( − D ) = n X i =1 O P ( α i ) with α ≤ α ≤ · · · ≤ α n and let e = gcd( r + 1 , n ) . Then: r n X i = n − r +1 α i − n n X i =1 α i ≤ (cid:18) r ( n − r ) + ( e − nr (cid:19) d − ( e − nr = ( n − r ) d n + ( e − d − nr . Remark 2.9.
In the extreme cases of rank one sub- or quotient bundles,these inequalities read:0 ≤ nα n − n X i =1 α i ≤ (cid:18) n − (cid:19) d, if n is odd; (cid:18) nd (cid:19) − , if n is even.Moreover, 0 ≤ n X i =1 α i − nα ≤ ( n − d − . Adding the two inequalities together gives a bound for α n − α on the orderof 3 d/ Lemma 2.10. (i)
Suppose that h ( O Y ( D )) = 0 . Then there is a non-split extension → O Y → V → O Y ( D − K Y ) → . Moreover c ( V ) = D − K Y and c ( V ) = 0 . (ii) Suppose that x is a base point of the linear system | D | . Then thereexists a rank two vector bundle V and an exact sequence → O Y → V → O Y ( D − K Y ) ⊗ m x → . In particular c ( V ) = D − K Y and c ( V ) = 1 . roof. (i) Obvious since h ( − ( D − K Y )) = h ( − D + K Y ) = h ( D ).(ii) The exact sequenceExt ( O Y ( D − K Y ) ⊗ m x , O Y ) α −→ H ( Ext ( O Y ( D − K Y ) ⊗ m x , O Y )) →→ H ( Y ; O Y ( − D + K Y )) → Ext ( O Y ( D − K Y ) ⊗ m x , O Y )is Serre dual to H ( Y ; O Y ( D ) ⊗ m x ) → H ( Y ; O Y ( D )) β −→ H ( k x ) → H ( Y ; O Y ( D ) ⊗ m x ) . Thus a locally free extension V exists ⇐⇒ α is nonzero ⇐⇒ α is surjective ⇐⇒ β = 0 ⇐⇒ x is a base point of | D | . Lemma 2.11. (i)
With V as in (i) of Lemma 2.10, if T is a divisorclass on Y and there exists a nonzero homomorphism O Y ( T ) → V ,then deg F T ≤ . In particular, the restriction of V to the genericfiber Y K is stable. (ii) With x and V as in (ii) of Lemma 2.10, if T is a divisor class on Y andthere exists a nonzero homomorphism O Y ( T ) → V , then deg F T ≤ .In particular, the restriction of V to the generic fiber Y K is stable.Proof. We shall just prove (ii), as the proof of (i) is simpler. Given O Y ( T ) → V , we may assume that the cokernel V / O Y ( T ) is torsion free. If the imageof O Y ( T ) is contained in the subsheaf O Y , then T = − E for some effectivedivisor E . In this case deg F T = − deg F E ≤
0. Otherwise the inducedhomomorphism O Y ( T ) → O Y ( D − K Y ) ⊗ m x is nonzero. Thus T = D − K Y − E for some effective divisor E with x ∈ Supp E . If deg F T >
0, thensince deg F T = deg F D − deg F E ≤ n , and the minimality of n , we musthave deg F E = 0, and, since E is effective, E = P i C i is a sum of irreduciblecurves C i contained in fibers of f . On the other hand, since V / O Y ( T ) istorsion free, there is an exact sequence0 → O Y ( T ) → V → O Y ( D − K Y − T ) ⊗ I Z → , where Z is a zero-dimensional subscheme of Y . Plugging in T = D − K Y − E ,we see that there is an exact sequence0 → O Y ( D − K Y − E ) → V → O Y ( E ) ⊗ I Z → . Hence c ( V ) = ( D − K Y − E ) · E + ℓ ( Z ). Since E is supported in the fibersof f , E · K Y = 0, E ≤ E = 0 ⇐⇒ E is numerically equivalent15o a multiple mF for some positive integer m . First suppose that E < E ≤ − E · K Y = 0. By our assumption, D · E ≥
0, and K Y · E = 0, so that c ( V ) ≥ − E ≥
2, which contradicts c ( V ) = 1. So E isnumerically equivalent to mF . In this case c ( V ) = ( D − K Y − E ) · E + ℓ ( Z ) = m ( D · E ) + ℓ ( Z ) ≥ mn ≥
2, which again contradicts c ( V ) = 1. Hencedeg F T ≤ Remark 2.12.
One can generalize the above as follows. Let Z be a zero-dimensional subscheme local complete intersection of Y ; for simplicity as-sume that Z = { p , . . . , p k } consists of k distinct points. Then a locally freeextension of the form0 → O Y → V → O Y ( D − K Y ) ⊗ I Z → ⇐⇒ Z has the Cayley-Bacharach property with respect to O Y ( D ):every section of O Y ( D ) which vanishes at all but one of the points p i vanishesat all of the points.Assume that a locally free extension V exists, and further assume forsimplicity that all fibers of f are irreducible. Let m = { f ( p ) , . . . , f ( p k ) } .Hence, if F , . . . , F r are distinct fibers of f and if Z ⊆ F + · · · + F r , then r ≥ m . Arguments as above show that, if mn ≥ k + 1, then every locallyfree V as above restricts to a stable bundle on the generic fiber. Corollary 2.13.
Suppose that n = deg F D is odd. (i) If h ( D ) = 0 , then D ≤ n ( d + 2 g − − d = (2 n − d + 4 n ( g − . (ii) If x is a base point of the linear system | D | , then D ≤ n ( d + 2 g − − d + 4 = (2 n − d + 4 n ( g −
1) + 4 . Proof.
Again, we shall just prove (ii). If such an x exists, then we haveconstructed a rank two bundle V with c ( V ) = D − K Y and c ( V ) = 1 suchthat the restriction of V to the generic fiber is semistable. The hypothesesof Theorem 1.2 apply, with e = 1. Thus4 c ( V ) − c ( V ) = 4 − D + 2 n ( d + 2 g − ≥ d, which yields the inequality D ≤ n ( d + 2 g − − d + 4 = (2 n − d + 4 n ( g −
1) + 4 . Corollary 2.14.
Suppose that n = deg F D is even and that char k = 2 . (i) If h ( D ) = 0 , then D ≤ (2 n − d + 4 n ( g − . If moreover B ∼ = P , then D ≤ (2 n − d − n − . (ii) If x is a base point of the linear system | D | , then D ≤ (2 n − d + 4 n ( g −
1) + 4 . If moreover B ∼ = P , then D ≤ (2 n − d − n + 2 . Let f : Y → B be a genus one fibration and let D be an f -nef divisor on Y of relative degree n >
1. Consider the P n − -bundle P = P ( f ∗ O Y ( D ) ∨ ) = P ( R f ∗ O Y ( − D )) over B (here our conventions are opposite to those of EGAor Hartshorne). Note that replacing D by D + tF leaves the projective spacebundle P unchanged. Denote by φ : P → B the projection and let P ⊆ P be the divisor class of a fiber. Let ¯ Y be the normal surface obtained bycontracting all of the curves which are orthogonal to D . There is a morphism g (of schemes over B ) from Y to P . For n ≥ g induces an embedding ¯ g of¯ Y . For n = 2, the corresponding morphism ¯ g : ¯ Y → P is a double cover. Byconstruction, g ∗ O P (1) = O Y ( D ), and hence g ∗ [ O P (1) ⊗ φ ∗ λ ] = O Y ( D ) ⊗ f ∗ λ . Lemma 3.1.
For all line bundles λ on B and all i ≥ , g ∗ induces anisomorphism H i ( P ; O P (1) ⊗ φ ∗ λ ) ∼ = H i ( Y ; O Y ( D ) ⊗ f ∗ λ ) . Proof.
First assume that Y = ¯ Y and hence D | E is ample for every fiber E of f . First suppose that n ≥
3, so that g is an embedding. There is an exactsequence 0 → O P (1) ⊗ I Y → O P (1) → O Y ( D ) → ,
17o it suffices to show that H i ( P ; O P (1) ⊗ I Y ⊗ φ ∗ λ ) = 0 for all i . Via Leray,it suffices to show that R i φ ∗ ( O P (1) ⊗ I Y ) = 0 for all i . Since O P (1) ⊗ I Y is flat over B , it suffices by base change to show that, for every fiber P of φ , H i ( P ; O P (1) ⊗ I Y ⊗ O P ) = 0. It is easy to check that, if E = P ∩ Y isthe corresponding fiber of f , then O P (1) ⊗ I Y ⊗ O P = O P n − (1) ⊗ I E . Thenthere is the corresponding exact sequence0 → O P n − (1) ⊗ I E → O P n − (1) → O E ( D ) → . For i > H i ( P n − ; O P n − (1)) = H i ( E ; O E ( D )) = 0, and for i = 0 therestriction homomorphism H ( P n − ; O P n − (1)) → H ( E ; O E ( D )) is an iso-morphism. It follows that H i ( P n − ; O P n − (1) ⊗ I E ) = 0 for every i . Thus R i φ ∗ O P (1) ⊗ I Y = 0 for all i and we are done in the case n ≥ n = 2, Y is a double cover of P branched along a section of O P (4) ⊗ φ ∗ µ ⊗ for some line bundle µ on B . Let L = O P (2) ⊗ φ ∗ µ . Then g ∗ O Y ( D ) = O P (1) ⊕ [ O P (1) ⊗ L − ] . So it is enough to show that H i ( P ; O P (1) ⊗ L − ⊗ φ ∗ λ ) = 0 for all i . Since O P (1) ⊗ L − restricts to O P ( −
1) on every fiber of φ , R i φ ∗ O P (1) ⊗ L − = 0for all i and thus H i ( P ; O P (1) ⊗ L − ⊗ φ ∗ λ ) = 0 as well.Finally, in case Y = ¯ Y , let ρ : Y → ¯ Y be the natural morphism. Then ρ is a resolution of singularities. Note that ¯ Y has rational singularities and D induces a relatively ample Cartier divisor ¯ D on ¯ Y with ρ ∗ ¯ D = D . The abovearguments show that H i ( P ; O P (1) ⊗ φ ∗ λ ) ∼ = H i ( ¯ Y ; O ¯ Y ( ¯ D ) ⊗ ¯ f ∗ λ ), where¯ f : ¯ Y → B is the induced morphism. Again using the fact that ¯ Y has rationalsingularities, R ρ ∗ ( O Y ( D ) ⊗ f ∗ λ ) = O ¯ Y ( ¯ D ) ⊗ ¯ f ∗ λ and R i ρ ∗ ( O Y ( D ) ⊗ f ∗ λ ) =0 for i >
0, so we are done via Leray again.For the rest of this section, we assume that B ∼ = P and that f ∗ O Y ( D ) = L ni =1 O P ( − α i ), where α ≤ α ≤ · · · ≤ α n . By convention, we choose theline bundle O P (1) such that φ ∗ O P (1) = L ni =1 O P ( − α i ). Lemma 3.2.
In the above notation, H ( P ; O P (1) ⊗ O P ( tP )) = 0 ⇐⇒ t ≤ α n − . Hence, H ( Y ; O Y ( D + tF )) = 0 ⇐⇒ t ≤ α n − .Proof. Immediate from the Leray spectral sequence and Lemma 3.1.Next let us describe linear systems on P = P ( L ni =1 O P ( α i )). For 1 ≤ i ≤ n , the summand O P ( α i ) gives a section Σ i of O P (1) ⊗ O P ( α i P ). Inparticular Σ is a section of O P (1) ⊗ O P ( α P ). Of course, the summand isnot unique except in the case i = 1 and α > α . Suppose that α = α = · · · = α i < α i +1 = α i +2 = · · · = α i < · · · < α i k +1 = · · · = α n . Considersections of the line bundle O P (1) ⊗ O P ( tP ). By direct inspection,18 emma 3.3. In the above notation, if α i j ≤ t < α i j +1 , then the base locus ofthe complete linear system corresponding to the line bundle O P (1) ⊗ O P ( tP ) is Σ ∩ Σ ∩ · · · ∩ Σ i j . In particular, this base locus is empty ⇐⇒ t ≥ α i k +1 = α n . Corollary 3.4.
Suppose that n > . The complete linear system | D + tF | has a nonempty base locus ⇐⇒ t ≤ α n − and, for the unique j such that α i j ≤ t < α i j +1 , Y ∩ Σ ∩ Σ ∩ · · · ∩ Σ i j = ∅ .If n = 2 , the complete linear system | D + tF | has a nonempty base locus ⇐⇒ t ≤ α − . We now look at examples for small values of n . The case n = 2 : Let Y be a genus one fibration of order 2. For simplicity,we shall just look at the case Y = ¯ Y , i.e. D is f -ample. We can normalize D so that the rank two vector bundle f ∗ O Y ( D ) is equal to O P ⊕ O P ( − a ),for some a ≥
0, and hence P = F a . There is thus a degree two morphism g : Y → F a . With this normalization, D = g ∗ σ , where σ is the negativesection (or σ = 0 if a = 0). Our bounds in the previous section all give thesingle inequality a ≤ d −
1. If B ⊆ F a is the branch divisor of f , then B =4 σ +2 N P where P is the class of a fiber of the ruled surface. Since Y has nosection B is smooth and irreducible with N ≥ a . But N = 2 a is impossiblesince then B ∩ σ = ∅ and the inverse image of σ on Y would consist of twosections. Thus N ≥ a + 1. Note that g ∗ O Y = O F a ⊕ O F a ( − σ − N P ). Thecanonical divisor K Y = g ∗ ( − σ − ( a + 2) P + 2 σ + N P ) = g ∗ ( N − a − P .Thus d = N − a . If N = 2 a + ε , then d = a + ε with ε ≥
1; thus we seedirectly that a ≤ d −
1. Finally, H ( Y ; O Y ( D + tF )) = 0 ⇐⇒ t ≤ a − | D + tF | has a base locus ⇐⇒ t ≤ a − Proposition 3.5.
Let k be uncountable. Let σ be the negative section of F a and let P be the class of a fiber. Suppose that a > and ε ≥ or that a = 0 and ε ≥ . Then, if B a generic element of | σ + 2(2 a + ε ) P | , thecorresponding double cover Y of F a has no section.Proof. Let N = 2 a + ε . First note that, for all N ≥ a , the linear system | σ + 2 N P | is base point free on F a . We consider degenerations X of F a to F a ∪ F , glued along a fiber P , obtained by taking a trivial family F a × A andblowing up a fiber of F a × { } in the threefold F a × A . For the fiber F a ∪ F over 0, the line bundle L = π ∗ O F a (4 σ + 2 N P ) ⊗ O F a × A ( − F ) restrictson the copy of F a to O F a (4 σ + 2( N − P ) and on the F component to19 F (4 σ + 2 P ). Clearly L is divisible by 2 in Pic X . If B is a smooth sectionof L , the double cover Y of X branched along B fibers over A . The generalfiber is a genus one fibration Y → P with χ ( O Y ) = N − a = a + ε and thefiber over 0 is of the form Y ∪ R , where Y → P is a genus one fibrationwith χ ( O Y ) = N − a − a + ε − R is a rational elliptic surface, and Y and R meet transversally along a smooth fiber F . If either a ≥ ε ≥ a = 0 and ε ≥ χ ( O Y ) ≥ Y is discrete. Ifthere is a divisor of fiber degree one on the general fiber (or equivalentlya section), then, possibly after a base change, there is a Cartier divisor on Y ∪ R which restricts to a section of both Y and R . If we have to make abase change of order k around 0 ∈ U , then the threefold Y acquires a curveof A k singularities and is resolved by a smooth threefold e Y with central fiber Y ∪ E ∪ · · · ∪ E k ∪ R , where the E i are elliptic ruled, E i ∩ E i +1 is a sectionin E i and in E i +1 , E ∩ Y = F and E k ∩ R = F . The argument in thegeneral case is then essentially reduced to the case e Y = Y and we shall justwrite down this case. There would thus be a divisor σ on Y and σ R on R , with σ · F = σ R · F = 1, and, for the common fiber F of Y and R , O Y ( σ ) | F = O R ( σ R ) | F . We shall show that, for generic choices, this isimpossible.For a fixed genus one fibration Y , the image of Pic Y in Pic F is afinitely generated abelian group Γ. On the other hand, fixing the ellipticcurve F , it is easy to check the following: for every choice p , . . . , p ofpoints p i ∈ F , there exists a rational elliptic surface R and an inclusionof F as a fiber of R , such that the image of Pic R in Pic F is generatedby the line bundles O F ( p i ), together with the image of the line bundle h ,which is a cube root of O F ( P i p i ). (Embed F in P by a linear system h such that 3 h ≡ P i p i and let R be the surface which is P blown up atthe images of the p i .) It is then a straightforward exercise to see that, aslong as k is uncountable, for a fixed finitely generated subgroup Γ of Pic F and for generic choices of the p , . . . , p , the intersection of the subgroupgenerated by the O F ( p i ) and h with Γ is trivial. In particular the equality O Y ( σ ) | F = O R ( σ R ) | F is impossible.We must still show that we can find a sufficiently general rational ellipticsurface as one component of the special fiber of a Y → A as constructedabove. View Y as a double cover of X as above. Fix a smooth divisor B in the linear system | σ + 2( N − P | on F a which meets a given fiber f transversally in 4 points; this is possible for all N ≥ a + 1. Given a smoothdivisor B in | σ + 2 P | on F such that B ∩ P = B ∩ P , the curve B ∪ B corresponds to a section s of L|X . A straightforward argument shows that H ( X ; L|X ) = 0, and hence that there is a Zariski open neighborhood U
20f 0 ∈ A such that the section s of L|X extends to a section of L over U (which we may assume to be smooth). The corresponding double coveris the degeneration we seek, provided that we can choose the section B sothat the rational elliptic surface R is sufficiently generic.To see that we can find a generic section B , it is sufficient to show that,if R is a generic rational elliptic surface containing a fiber isomorphic to F ,then R is a double cover of F , necessarily branched along a smooth curve in | σ + 2 P | (possibly after switching the order of the factors of F ∼ = P × P ).It is enough to consider the case where R is any rational elliptic surface withall fibers irreducible, or equivalently such that there does not exist a smoothrational curve on R with self-intersection −
2. In this case, we claim thatthere is a section τ of R such that τ · σ R = 1. It is then easy to check thatthe linear system | σ R + τ + F | has dimension 3 and degree 4, and definesa morphism ϕ : R → P which is of degree 2 onto its image, which is asmooth quadric in P . Thus ϕ realizes R as a double cover of F . To find τ ,use the fact that (as all fibers are irreducible) sections of R correspond toelements of { σ R , F } ⊥ ∼ = − E ⊆ Pic R , and that the condition that a section τ corresponding to α ∈ − E satisfy σ R · τ = 1 is α = −
4. Since there doexist vectors in − E of square −
4, we can find R as desired. This completesthe proof.The following corollary shows that Corollary 2.8 and Corollary 2.14 aresharp in the case of order 2: Corollary 3.6.
Let d be an integer, d ≥ . (i) For all a , ≤ a ≤ d − , there exists a genus one fibration Y → P oforder which is a double cover of F a . (ii) There exist a genus one fibration Y → P of order and a divisor D on Y such that D · F = 2 , h ( D ) = 0 and D = 2 d − . (iii) There exist a genus one fibration Y → P of order and a divisor D on Y such that D · F = 2 , | D | has a base point and D = 2 d − .Proof. Given a , 0 ≤ a ≤ d −
1, take Y to be a generic double cover of F a branched along a section of 4 σ + 2(2 a + ε ) P , where ε = d − a . Note that,if a = 0, then ε ≥
2. The induced morphism f : Y → P realizes Y asa genus one fibration (clearly without multiple fibers) with a divisor class D = g ∗ σ such that D · F = 2. By Proposition 3.5, Y has order 2, andthe discussion before the proof of Proposition 3.5 shows that χ ( O Y ) = d .This proves (i). Similarly, with Y as in (i) and with D = g ∗ σ , consider21 + ( a − F = g ∗ ( σ + ( a − P ). Then h ( D + ( a − F ) = 0 and D = − a + 4( a −
2) = 2 a − d −
10. This proves (ii), and (iii) followssimilarly by considering D + ( a − F . The case n = 3 : Let Y be a genus one fibration of order 3. Again, we shalljust look at the case Y = ¯ Y , i.e. D is f -ample. We can normalize D so thatthe rank two vector bundle f ∗ O Y ( D ) is equal to O P ⊕ O P ( − a ) ⊕ O P ( − b ),where 0 ≤ a ≤ b . The bounds of Corollary 2.8 give 2 b − a ≤ d and a + b ≤ d − P = P ( O P ⊕ O P ( a ) ⊕ O P ( b )) with φ : P → P the projection. Thereis a surface Σ ⊆ P with φ | Σ a P -bundle whose fibers are linearly embeddedin the fibers of φ and such that φ ∗ O P (Σ ) = O P ⊕ O P ( − a ) ⊕ O P ( − b ), andit is unique if a >
0. By our assumptions, g : Y → P embeds Y as a smoothsurface in the linear system | + N P | , and we may choose D so that D = g ∗ Σ as divisor classes. Lemma 3.7.
In the above notation, d = N − ( a + b ) . Hence N ≥ b , and,if a = b , N ≥ b + 1 .Proof. First, we have the following formula for the relative dualizing sheaf: ω P / P = O P ( − ) ⊗ φ ∗ det( O P ⊕O P ( − a ) ⊕O P ( − b )) = O P ( − − ( a + b ) P ) . Thus K P = O P ( − − ( a + b + 2) P ). By adjunction K Y = O Y (( N − ( a + b + 2)) F and thus d = N − ( a + b ).To see the last statement, by Corollary 2.8 N = a + b + d ≥ a + b +(2 b − a ) = 3 b . If a = b , then a + b = 2 b ≤ d −
2, so that d ≥ b + 1 and N = a + b + d ≥ b + 1. Lemma 3.8. If D t = (Σ + tP ) | Y = Σ | Y + tF , then: (i) D t = − a − b + d + 6 t . (ii) H ( Y ; O Y ( D t )) = 0 ⇐⇒ t ≤ b − . Hence H ( Y ; O Y ( D )) = 0 ⇐⇒ D ≤ d − a + 4 b − . (iii) If N ≥ b + 1 and a < b , the linear system | D t | has a base locus forall t ≤ b − .Proof. We begin with the following claim:
Claim 3.9.
With notation as above, Σ ∼ = F b − a and, if σ denotes thenegative section of Σ , then O Σ (Σ ) = O Σ ( σ − af ) . roof of the claim. Applying φ ∗ to the exact sequence0 → O P → O P (Σ ) → O Σ (Σ ) → , we get 0 → O P → O P ⊕ O P ( − a ) ⊕ O P ( − b ) → φ ∗ O Σ (Σ ) → , so that φ ∗ O Σ (Σ ) = O P ( − a ) ⊕ O P ( − b ) and Σ = F b − a . If σ is thenegative section of Σ , then φ ∗ O Σ ( σ ) = O P ⊕ O P ( a − b ). Since O Σ (Σ )has degree one on each fiber of Σ , O Σ (Σ ) = O Σ ( σ − af ) . Returning to the proof of Lemma 3.8, to see (i) it is enough to consider D = D . In this case D = Σ · Σ · (3Σ + N P ) = ( σ − af ) · (3 σ +( N − a ) f ),where this last intersection is computed on Σ . Thus D = 3( a − b ) − a + N − a = − a − b + N = − a − b + d, establishing (i). Part (ii) then follows from Lemma 3.2.Finally, to see (iii), note that, for t < b , the negative section σ of Σ iscontained in the base locus of | Σ + tP | since (Σ + tP ) | Σ = σ + ( t − a ) f .Now Y · σ = (3 σ + ( N − a ) f ) · σ = 3 a − b + N − a = N − b. Thus, since N ≥ b + 1, Y ∩ σ = ∅ and hence the base locus of | D + tF | contains the nonempty set Y ∩ σ , by Lemma 3.1. This completes the proofof (iii). Remark 3.10.
In the situation of (iii) above, if N = 3 b , then Y · σ = 0.Thus either Y ∩ σ = ∅ or σ ⊆ Y . If Y ∩ σ = ∅ , as long as t > a , it is easyto check that | D + tF | is base point free. If σ ⊆ Y , then σ is a section of Y and hence Y does not have order 2.As in the case n = 2, we now construct examples of Y of order 3: Proposition 3.11. (i)
For N ≥ b − , the restriction homomorphism H ( P ; O P (3Σ + N P )) → H (Σ ; O Σ (3Σ + N P )) is surjective. Suppose that b > a . The base locus of | + (3 b − P | is σ . For N = 3 b − , there exist smooth surfaces Y in the linear system | +(3 b − P | . Every such Y contains σ , which is a section of the genusone fibration. (iii) For N ≥ b , the linear system | + N P | is base point free and hencecontains smooth surfaces Y . If moreover b > a and k is uncountable,the generic surface Y in | + N P | satisfies: every line bundle L on Y has deg F L ≡ , where F is the divisor class P | Y . (iv) Suppose a = b and k is uncountable. Further suppose either that b > and N ≥ b + 1 or that b = 0 and N ≥ . Then the generic surface Y in | + N P | satisfies: every line bundle L on Y has deg F L ≡ .Proof. To prove (i), it suffices to show that H ( P ; O P (2Σ + N P )) = 0. Now R φ ∗ O P (2Σ + N P ) = 0 and R φ ∗ O P (2Σ + N P ) = Sym ( O P ⊕ O P ( − a ) ⊕ O P ( − b )) ⊗ O P ( N ) . The most negative term in the direct sum is O P ( − b + N ). Thus H ( P ; R φ ∗ O P (2Σ + N P )) = 0as long as N ≥ b −
1, and (i) follows from the Leray spectral sequence.Clearly 3 b − ≥ b −
1. Note that | + N P | Σ = 3 σ − af + N f ,which is base point free ⇐⇒ N ≥ b . Thus, by (i), if N ≥ b , then | + N P | is base point free and hence the generic Y ∈ | + N P | issmooth. This proves the first sentence in (iii). For (ii), 3 σ − af +(3 b − f = σ + (2 σ + (3 b − a − f ). This is of the form σ + base point free linearsystem, since 3 b − a − ≥ b − a (because b − a ≥ Y is smooth one can make a local computation along the base locus;details omitted.To see the second statement of (iii), we must show that, for N ≥ b and b > a , the generic Y has no divisor whose fiber degree is positive and lessthan 3, or equivalently the genus one fibration on Y has no section. Theproof will be via degenerations, beginning with the following construction:Let Y ∈ | + ( N − P | be a smooth surface, guaranteed to exist by (ii)and the first part of (iii) for all N ≥ b . Note that χ ( O Y ) = N − − a − b ≥ b − − a − b = b + ( b − a ) − ≥ b > . Y is discrete. Choose a smooth fiber F on Y and let P be thefiber of P corresponding to F . There is the normal crossings surface Y ∪ P ,whose double curve is F , a cubic in P . Let Y be a smooth surface in | + N P | . Then we can make a threefold ¯ Z corresponding to the totalspace of the pencil spanned by Y and Y ∪ P . The threefold ¯ Z is singularexactly at the intersection Y ∩ F . If this intersection is transverse, thenthe singularities of ¯ Z will be nine threefold ordinary double points, since Y ∈ | + N P | meets the P in a plane cubic. As is well known, there is asmall projective resolution Z of ¯ Z whose central fiber is Y ∪ R , where R isthe rational elliptic surface obtained by blowing up P along the nine points Y ∩ F . We next show that, for N ≥ b , the nine points on F are arbitrarysubject to the condition that they are cut out by a plane cubic: Claim 3.12. If N ≥ b , then the restriction map H ( P ; O P (3Σ + N P )) → H ( P ; O P (3)) is surjective.Proof. We show that H ( P ; O P (3Σ + ( N − F )) = 0. As usual, applyLeray. Clearly R φ ∗ O P (3Σ + ( N − F ) = 0. Moreover, R φ ∗ O P (3Σ + ( N − F ) = Sym ( O P ⊕ O P ( − a ) ⊕ O P ( − b )) ⊗ O P ( N − . The most negative term that appears is O P ( − b + N − H ( R φ ∗ O P (3Σ + ( N − F )) = 0, it suffices to take − b + N − ≥ − N ≥ b .Returning to the proof of the lemma, we may assume that R is theblowup of P at nine general points p , . . . , p ∈ F , subject to the conditionthat p + · · · + p ∈ |O P (3) | F | . Equivalently, we can choose the eight points p , . . . , p arbitrarily and then the ninth is determined.Replacing the base P of the pencil by a Zariski open subset ∆, we mayassume that f : Z → ∆ is projective, with all fibers smooth except the fiberover 0, which is Y ∪ R . If for all t the fiber f − ( t ) has a section, then perhapsafter a base change there is a line bundle L over Z such that L| f − ( t ) hasfiber degree one. The argument in case we need to make a base change isthen essentially reduced to the case e Z = Z (as in our discussion of the case n = 2) and we shall just write down this case.Consider L| ( Y ∪ R ). This corresponds to line bundles L on Y and L on R whose restrictions to F agree. Let G be the finitely generated subgroupof Pic F which is the image of the restriction map Pic Y → Pic F . Then25here is an equation in Div F of the form q ≡ ah + X i =1 n i p i . Here q ∈ F corresponds to some element in G of degree one, h ∈ |O P (3) | F | ,and the n i ∈ Z . Since h has degree 3 and q has degree one, the n i are not all0. Equivalently, P i =1 n i p i is a nonzero element lying in some fixed finitelygenerated subgroup of F . But since F is uncountable, it is clearly possibleto choose the p i so that this does not happen. This is a contradiction. Hencethe generic Y has no section.The proof of (iv) is similar. Remark 3.13.
In the situation of (iv), with a = b and N = 3 b , Y ∈| + 3 bF | , Σ ∼ = F , and Y · F = 3 σ . Since every element in | σ | is asum of three elements in | σ | , Y contains a section.We now show that Corollary 2.8 and Corollary 2.13 are sharp in the caseof order 3. Corollary 3.14.
Let d be an integer, d ≥ . (i) Let a and b be two integers, ≤ a ≤ b , such that b − a ≤ d and a + b ≤ d − . There exist a genus one fibration Y → P with χ ( O Y ) = d of order which is embedded in P = P ( O P ⊕ O P ( a ) ⊕ O P ( b )) as adivisor of relative degree . (ii) There exist a genus one fibration Y → P with χ ( O Y ) = d of order and a divisor D on Y with D · F = 3 , h ( D ) = 0 and D = 3 d − . (iii) Suppose that d ≥ . There exist a genus one fibration Y → P with χ ( O Y ) = d of order and a divisor D on Y with D · F = 3 , such that | D | has a base point, and D = 3 d − .Proof. To see (i), first suppose that a > b . In this case, since 2 b − a = b +( b − a ) ≤ d , b ≤ d −
1, and hence the inequality a + b ≤ d − b − a ≤ d . Set N = a + b + d ≥ ( a + b ) + (2 b − a ) = 3 b .By (iii) of Proposition 3.11, there exists a smooth Y ∈ | + N P | with nosection and with χ ( O Y ) = d . The projection P → P realizes Y as a genusone fibration (with no multiple fibers), and D = Σ | Y satisfies D · F = 3.Thus Y has order 3.In case a = b , the inequalities 2 b − a ≤ d and a + b ≤ d − b ≤ d −
1. In this case set N = a + b + d ≥ b + b + 1 = 3 b + 1, and N = d ≥ a = b = 0. The argument then concludes as in the case a < b but using(iv) of Proposition 3.11.To see (ii), choose a < b such that d = 2 b − a , which is possible as longas d ≥
2, Let Y be as in (i), and consider the divisor D = Σ | Y + ( b − F = D b − , in the notation of (ii) of Lemma 3.8, then h ( D ) = 0 and D = d + 4 b − a −
12 = 3 d −
12. Finally, to see (iii), choose a < b such that d = 2 b − a + 1 and let Y be as in (i). Then N = a + b + d = 3 b + 1. By (iii)of Lemma 3.8, if D = Σ | Y + ( b − F = D b − , the linear system | D | has abase point, and D = d + 4 b − a − d − Remark 3.15.
In case d = 2 in (iii) above, Y is a K D with D effective, D · F = 3 and D = − | D | has a fixed component. It suffices to take a = b = 1, N = 4, and D = Σ · Y . Let k be algebraically closed and let n be a positive integer prime to thecharacteristic of k . Later, in the discussion of Pontrjagin squares, we willfurther have to assume that char k = 2, and eventually that k = C . Ourgoal in this section is to describe the set of components of the moduli spaceof genus one fibrations over P . We shall primarily be concerned with thecase where all fibers of π are irreducible, because of the following: Theorem 4.1.
Suppose that k = C . Let f : Y → P be a genus one fibrationand let D be a divisor on Y of fiber degree n . Then there exists a complexmanifold Y , a flat proper morphism Φ :
Y → P × ∆ , where ∆ is the unitdisk in C , and a divisor D on Y such that, if Ψ :
Y → ∆ is the composition,then Ψ − (0) = Y = Y , Ψ − ( t ) = Y t is a smooth genus one fibration over P with all fibers irreducible for all t = 0 , and D induces a divisor on Y t forall t , such that D| Y has the same restriction to the generic fiber as D .Proof. (Sketch.) The proof of this theorem uses standard results going backto Kodaira (see also Kas [15] or [13, Section 1.5] for more details). Let M d bethe coarse moduli space of elliptic surfaces π : X → P with χ ( X ; O X ) = d .(One has to be a little careful in case d = 1.) Then M d is irreducible, sincevia the existence of Weierstrass models there is a surjection U → M d , where U is a nonempty Zariski open subset in the product |O P (4 d ) | × |O P (6 d ) | .In particular, beginning with an elliptic surface π : X → P , there exists asmooth curve, which we may take to be the unit disk ∆, and an ellipticfibration with a section X → ∆ × P such that, if Π : X → ∆ is the induced27ap, then Π − (0) ∼ = X and Π − ( s ) is an elliptic surface all of whose fibersare irreducible for s = 0. Here, one needs the existence of simultaneousresolution of surface rational double points since the Weierstrass model ofan elliptic surface with reducible fibers will be singular.If π : X → B is an elliptic surface, there is an analytic Tate-Shafarevichgroup H ( B ; E ) and a surjective homomorphism H ( X ; O X ) → H ( B ; E )[13, Section 1.5, Lemma 5.11]. Let Π : X → S be an arbitrary family ofelliptic surfaces over a complex manifold S . Then a point of the total space V = V ( R Π ∗ O X ) corresponds to a complex elliptic surface Y s whose Jaco-bian surface is isomorphic to X s = Π − ( s ). Moreover, there is a tautologicalfamily of genus one fibrations Ψ : Y → V [13, Section 1.5, Proposition 5.31].Specializing to the case B = P , let f : Y → P and D be as in the statementof the theorem, let X be the Jacobian surface of Y , and let X → ∆ be aone-parameter deformation of X as in the preceding paragraph. Thus thereis a point v ∈ V = V ( R Π ∗ O X ) corresponding to Y , and a tautologicalfamily Ψ : Y → V . Let ξ ∈ H ( Y ; Z ) be the class [ D ]. Then, as V is simplyconnected, ξ extends to a section σ of R Ψ ∗ Z over V , and the locus Ξ where ξ is of type (1 ,
1) corresponds to the set of points v ∈ V where the image of σ in the fiber of R Ψ ∗ O Y over v is 0. Hence Ξ is an analytic subset of V andevery component of Ξ has codimension at most h , ( Y ) = h , ( X ) = d − H ( X ; O X )is countable. Thus, Ξ ∩ V (0) is discrete and the induced morphism Ξ → ∆is nonconstant on every component. Choosing the normalization of a com-ponent of Ξ passing through v , which we can assume is isomorphic to adisk, and replacing Ψ by its restriction to this component, then gives adeformation of Y as desired.For arbitrary fields k , we can only show that Y can be deformed toa genus one fibration Y t over P with all fibers irreducible and such thatthere exists a divisor on Y t whose fiber degree is divisible by n . However,a stronger result is most likely true in positive characteristic as well. Inwhat follows, we assume that all fibers of π are irreducible unlessexplicitly stated. More generally let π : X → B be an elliptic surface (with a fixed section).The set of triples ( Y, f, ϕ ), where f : Y → B is a genus one fibration and ϕ : J ( Y ) → X is an isomorphism from the Jacobian surface of Y to X over B sending the zero section of J ( Y ) to the fixed section of X , can be identifiedwith the group H ( B ; E ), where E is the sheaf of sections of π : X → B (andcohomology means ´etale cohomology). We denote the class of the triple28 Y, f, ϕ ) in H ( B ; E ) by [ Y, ϕ ]. There is a split exact sequence0 → E → R π ∗ G m / T deg −−→ Z → , where T is generated by the line bundles associated to components of fibersand the homomorphism R π ∗ G m / T → Z is essentially given by taking de-gree along the fibers. In particular, if all fibers of π are irreducible, thesheaf E may be identified with a subsheaf of R π ∗ G m . As B is a curve, theprincipal homogeneous space Y is trivial if and only if its restriction to thegeneric point Spec K of B is trivial (since a point over Spec K automati-cally extends to a section of f ). From this, it is easy to see that the point ξ ∈ H ( B ; E ) corresponding to ( Y, f, ϕ ) is an n -torsion point if and only ifthere exists a divisor D on Y with D · F = n . Thus the order of Y as definedin the introduction is the same as the order of the corresponding element of H ( B ; E ).There is an analogous construction for B an excellent Noetherian schemeand π : X → B a proper flat morphism, not necessarily smooth, all of whosegeometric fibers are irreducible of arithmetic genus one, with a section σ whose image is contained in the smooth locus of π (cf. for example [16],[2], [6]). Explicitly, if E is the smooth locus of π , then there is a canonicalidentification of group schemes E ∼ = Pic X /B which maps a section τ of E → B to the line bundle O X ( τ − σ ). A straightforward argument using theexistence of Weierstrass equations or the compactification of the relativePicard scheme ([1] or [5, Section 8.2]) shows that the action of E on itselfby multiplication extends to an action E × B X → X . Now suppose that f : Y → B is a proper flat morphism, all of whose geometric fibers areirreducible of arithmetic genus one, and that we are given an isomorphismof group schemes ϕ : Pic Y /B → E . Then, as shown in [2], if Y is the smoothlocus of f , then Y is a principal homogeneous space for E . Denote by[ Y , ϕ ] the corresponding element of H ( B ; E ). Given ϕ , we can construct acompactification Y ′ of Y by taking the associated space Y × E X which isthe quotient of Y × B X by the antidiagonal action of E . In the geometriccase, where B is a smooth curve over k , Y ∼ = Y ′ , since both surfaces haveunique relatively minimal models. In general, we shall always make theassumption that Y ∼ = Y ′ , i.e. that Y is determined by (and determines) theclass [ Y , ϕ ] ∈ H ( B ; E ), given the fixed compactification X of E .Suppose that n is a positive integer invertible on B . The n -torsion pointsin H ( B ; E ) can be described as follows: begin with the exact sequence (ofsheaves on X ) { } → µ n → G m × n −−→ G m → { } . π ∗ G m = G m , the homomorphism R π ∗ µ n → R π ∗ G m is injective.It follows that there is an injection R π ∗ µ n → E whose image is the ker-nel of multiplication by n , and hence a homomorphism H ( B ; R π ∗ µ n ) → H ( B ; E ) whose image is in the subgroup of n -torsion in H ( B ; E ). Thisimage is actually equal to the n -torsion because multiplication by n on E issurjective (see below).One can also proceed as follows: define E [ n ] via the exact sequence0 → E [ n ] → E × n −−→ E → , where we have used the assumption that the fibers of π are irreducibleto conclude that multiplication by n is surjective. The above argumentsshow that E [ n ] ∼ = R π ∗ µ n . Moreover there is an induced homomorphism H ( B ; E [ n ]) → H ( B ; E ) whose image is the subgroup of n -torsion pointsin H ( B ; E ). However the homomorphism H ( B ; E [ n ]) → H ( B ; E ) is notalways injective. Artin and Swinnerton-Dyer have proved ([2], Proposition(1.7)): Lemma 4.2.
The group H ( B ; E [ n ]) classifies the set of isomorphism classesof quadruples ( Y , f, ϕ, τ ) , where f : Y → B is a flat proper morphism, allof whose geometric fibers have arithmetic genus one, ϕ : Pic Y /B → E is anisomorphism of group schemes, with Y ∼ = Y × E X as schemes over B ,and τ is a section of Pic Y /B over B of degree n . Here an isomorphism Ψ : ( Y , f, ϕ, τ ) → ( Y ′ , f ′ , ϕ ′ , τ ′ ) consists of an isomorphism Ψ : Y → ( Y ′ ) such that f ′ ◦ Ψ = f , ϕ ′ ◦ Ψ ∗ = ϕ , where Ψ ∗ is the naturally induced isomor-phism from E to E ′ , and Ψ ∗ τ ′ = τ . Corollary 4.3.
Suppose that B is a smooth curve over k . Then the group H ( B ; E [ n ]) classifies the set of isomorphism classes of quadruples ( Y, f, ϕ, D ) ,where f : Y → B is a genus one fibration, ϕ is an isomorphism of ellipticsurfaces with a section from J ( Y ) to X , and D is a divisor on Y such that D · F = n , and an isomorphism Ψ : (
Y, f, ϕ, D ) → ( Y ′ , f ′ , ϕ ′ , D ′ ) consists ofan isomorphism Ψ : Y → Y ′ such that f ′ ◦ Ψ = f , ϕ ′ ◦ Ψ ∗ = ϕ , where Ψ ∗ isthe naturally induced isomorphism from J ( Y ) to J ( Y ′ ) , and Ψ ∗ D ′ has thesame restriction to the generic fiber of f as D . In the surface case, given a quadruple (
Y, f, ϕ, D ) as above, a concreteway to determine the class [
Y, ϕ, D ] = α ∈ H ( B ; E [ n ]) = H ( B ; R π ∗ µ n )given by Lemma 4.2 is via the following description of the filtration on H ( X ; µ n ) induced by the Leray spectral sequence. Let ρ : H ( X ; µ n ) → ( B ; R π ∗ µ n ) be the usual homomorphism. The pullback H ( B ; µ n ) → H ( X ; µ n ) has image equal to Z /n Z · [ F ], which is contained in Ker ρ , and H ( B ; R π ∗ µ n ) ∼ = Ker ρ/ ( Z /n Z · [ F ]) . As all fibers of π are irreducible, the homomorphism H ( X ; µ n ) → H ( B ; R π ∗ µ n ) ∼ = Z /n Z sends α to α ([ F ]), and hence H ( B ; R π ∗ µ n ) ∼ = [ F ] ⊥ / ( Z /n Z · [ F ]). Similarstatements hold when we replace π : X → B with f : Y → B . If D is adivisor on Y with D · F = n , then D defines a class [ D ] in H ( Y ; µ N ) forany N . Taking N = n , we see that the image of [ D ] in H ( B ; R f ∗ µ n ) is 0and hence D defines a class in H ( B ; R f ∗ µ n ), also denoted [ D ]. Lemma 4.4.
For every N prime to the characteristic and for all i , there isan isomorphism R i f ∗ µ N ∼ = R i π ∗ µ N which respects the cup product. Via the identification R f ∗ µ n ∼ = R π ∗ µ n ,the element − [ D ] ∈ H ( B ; R f ∗ µ n ) corresponds to the class α = [ Y, ϕ, D ] ∈ H ( B ; R π ∗ µ n ) given by Lemma 4.2.Proof. The first statement follows from the fact that there is an ´etale opencover of B , say { U i } , such that π − ( U i ) ∼ = f − ( U i ) as schemes over U i , andsuch that the transition functions are given by fiberwise translation and thusinduce canonical isomorphisms in cohomology. (This result does not needthe assumption that all fibers of π are irreducible.)To see the last statement, if { U i } is an ´etale open cover of B , we denotethe pullback of σ to U i by σ i and the pullback of σ to U i × B U j by σ ij . Theclass α = [ Y, ϕ, D ] given in [2, Proposition 1.7] is defined as follows: thegroup H ( B ; R π ∗ µ n ) = H ( B ; E [ n ]) is the first hypercohomology group H ( B ; E × n −−→ E ), and as such corresponds in ˇCech cohomology to a pair( ξ, δ ), where ξ = { ξ ij } is a 1-cocycle for E , δ = { δ i } is a 0-chain, and nξ ij = δ i − δ j . To pass from this description to an element of H ( B ; E [ n ]),write δ i = nγ i for some section γ i of X U i , which is possible after refiningthe cover, and then ξ ij − γ i + γ j is an n -torsion section of E over U i × B U j ,corresponding to the n -torsion line bundle O X Ui × BUj ( ξ ij − γ i + γ j − σ ij ) , which is then the corresponding representative for ( ξ, δ ). (Here and in whatfollows all equalities of line bundles are modulo Pic( U i ) or Pic( U i × B U j )31s appropriate.) Given Y and D , the class α = ( ξ, δ ) is defined as follows:We may assume that there exist sections τ i on Y U i . The isomorphism ϕ then uniquely defines an isomorphism of pairs ϕ i : ( Y U i , τ i ) → ( X U i , σ i ),and ϕ i ◦ ( ϕ j ) − is translation on X U i × B U j by a section ξ ij . We can write O Y Ui ( D ) = ϕ ∗ i O X Ui ( δ i + ( n − σ i ) for a unique section δ i . Thus( ϕ i ◦ ( ϕ − j )) ∗ ( δ i + ( n − σ ij )) = δ i + ( n − σ ij ) + nξ ij = δ j + ( n − σ ij ) , and so δ i − δ j = nξ ij on U i × B U j . After passing to a refinement of the cover,we can write δ i = nγ i as sections of E ( U i ); as divisors, δ i = nγ i − ( n − σ i ,and thus δ i + ( n − σ i = nγ i . It follows that ϕ ∗ i O X Ui ( γ i ) is an n th root of O Y Ui ( D ).On the other hand, the class [ D ] ∈ H ( Y ; µ n ) is the image of the linebundle O Y ( D ) ∈ H ( Y ; G m ) under the Kummer exact sequence. There isalso the exact sequence0 → R f ∗ µ n → R f ∗ G m → R → , where R is the image in R f ∗ G m of multiplication by n from R f ∗ G m to R f ∗ G m . Under the assumption that all fibers are irreducible, it is easyto see that O Y ( D ) defines a class in H ( B ; R ), i.e. that there exists an´etale open cover { U i } of B such that O Y ( D ) | Y U i = L ⊗ ni . So we can write O Y ( D ) | Y U i = L ⊗ ni , where L i has degree one on every fiber. The line bundle L i ⊗ L − j on Y U i × B U j is then a line bundle which is n -torsion on every fiberand so corresponds to a section of H ( Y U i × B U j ; µ n ). In this way we get aˇCech cocycle which defines an element of H ( B ; R f ∗ µ n ) corresponding to[ D ]. But as we have seen, we can take the n th root L i of O Y ( D ) | Y U i to bethe line bundle ( ϕ − i ) ∗ O X Ui ( γ i ) in the notation of the preceding paragraph.Hence the line bundle L i ⊗ L − j on Y U i × B U j corresponds to the line bundle O X Ui × BUj ( γ i − ( γ j + ξ ij − σ ij ) . Up to sign, this is the same as the class [
Y, ϕ, D ]. Remark 4.5.
One can show the following: suppose that π : X → B isa proper flat morphism with all fibers of π irreducible curves of arithmeticgenus one, where B is an excellent scheme of finite dimension and n is invert-ible in O B . Let f : Y → B is a proper flat morphism, all of whose geometricfibers are irreducible of arithmetic genus one, and that ϕ : Pic Y /B → E is32n isomorphism of group schemes. Suppose that the class [ Y , ϕ ] ∈ H ( B ; E )is divisible by n in the following sense: for every m ≥
1, there exists a ξ ∈ H ( B ; E ) such that [ Y , ϕ ] = n m ξ . Finally, let N be an integer invertiblein O B .Then R f ∗ µ N ∼ = R π ∗ µ N (respecting the cup product), which impliesboth that R i f ∗ µ N ∼ = R i π ∗ µ N , by taking cohomology sheaves, and that H i ( Y ; µ N ) ∼ = H i ( X ; µ N ), by taking hypercohomology. This is the analogueof the situation for k = C , where in fact X and Y are diffeomorphic.It is not clear if this result continues to hold if we drop the divisibilityassumption on the class of Y in H ( B ; E ), for example if B is a curve overa finite field.Using the exact sequence 0 → E [ n ] → E → E → E [ n ] ∼ = R π ∗ µ n , we see that the hypothesis on the divisibility of [ Y , ϕ ] issatisfied if H ( B ; R π ∗ µ n ) = 0. If for example π : X → B is a smoothelliptic surface with all fibers irreducible and at least one singular fiber,then one can check that H ( B ; R π ∗ µ n ) = 0 (cf. Remark 4.7) and hencethat R f ∗ µ n ∼ = R π ∗ µ n .Returning to the case of surfaces, we have the following results on thecohomology of Y , which are well-known if k = C : Proposition 4.6.
Let f : Y → B be a genus one fibration and let N be apositive integer prime to char k . (Note: we do not assume that all fibers of π are irreducible.) Suppose that f has a singular fiber. Then: (i) The induced homomorphisms H ( B ; µ N ) → H ( Y ; µ N ) and π ( Y, ∗ ) → π ( B, ∗ ) are isomorphisms. (ii) The group H ( Y ; µ N ) is a free Z /N Z -module. (iii) There exists a class θ ∈ H ( Y ; µ N ) such that θ · [ F ] = 1 , i.e. the imageof θ in H ( B ; R f ∗ µ N ) ∼ = Z /N Z is .Proof. (i) Let ν : Y ′ → Y be a finite ´etale cover, and take the Stein factor-ization of the composite morphism Y ′ → Y → B : Y ′ ν −−−−→ Y f ′ y y f B ′ −−−−→ B. Since f has no multiple fibers, B ′ → B is ´etale. Consider the inducedmorphism Y ′ → Y × B B ′ . Since Y ′ → Y and Y × B B ′ → Y are ´etale,33 ′ → Y × B B ′ is ´etale as well, and the statement of (i) is equivalent to theassertion that Y ′ → Y × B B ′ is an isomorphism. Replacing Y by Y × B B ′ ,which also has a singular fiber, it clearly suffices to prove that, if ν : Y ′ → Y is ´etale and the general fiber of the composite morphism f ′ = f ◦ ν : Y ′ → B is connected, then ν is an isomorphism. Note that, if F is a smooth fiber of f , then ν − ( F ) = F ′ is a connected ´etale cover of F , of degree ℓ , say. Thus Y ′ is again a genus one fibration over B , with no multiple fibers and withJacobian surface J ( Y ′ ) = X ′ . Since all fibers of f ′ are connected, the onlypossible singular fibers of f : Y → B are of type I k , and at least one suchfiber exists. Moreover, the fibers of f ′ are smooth, if they lie over smoothfibers of f , or of type I ℓk , if they lie over fibers of f of type I k . In particular,if χ ( Y ; O Y ) = d >
0, so that the Euler number e ( Y ) = c ( Y ) = 12 d , then e ( Y ′ ) = ℓe ( Y ) = 12 dℓ and hence χ ( Y ′ ; O Y ′ ) = dℓ .On the other hand, χ ( Y ′ ; O Y ′ ) = 1 − q ( Y ′ ) + p g ( Y ′ ). Since Y ′ hasno multiple fibers, R ( f ′ ) ∗ O Y ′ = L is a line bundle on B with deg L =deg( ω X ′ /B ) − <
0, since Y ′ and hence X ′ have a nonmultiple singular fiber.Thus H ( B ; R ( f ′ ) ∗ O Y ′ ) = 0 and it follows from the Leray spectral sequencethat q ( Y ′ ) = h ( Y ′ ; O Y ′ ) = g ( B ) = q ( Y ). Since ν : Y ′ → Y is ´etale, K Y ′ = ν ∗ K Y and hence p g ( Y ′ ) = h ( Y ′ ; K Y ′ ) = h ( Y ′ ; ν ∗ K Y ) = h ( Y ; ν ∗ ν ∗ K Y ).The homomorphism H ( Y ; K Y ) → H ( Y ; ν ∗ ν ∗ K Y ) is an isomorphism, be-cause the restriction of the quotient ν ∗ ν ∗ K Y /K K to a smooth fiber F is ofthe form L ℓ − i =1 λ − i , where λ is a torsion line bundle of order ℓ and hencehas no nonzero sections on F . Thus p g ( Y ′ ) = h ( Y ; K Y ) = p g ( Y ). But then χ ( Y ′ ; O Y ′ ) = 1 − q ( Y ′ ) + p g ( Y ′ ) = χ ( Y ; O Y ) = d , contradicting the fact that χ ( Y ′ ; O Y ′ ) = dℓ and that d > H ( Y ; µ N ) ∼ = H ( B ; µ N ) is a free Z /N Z -module, and hence H ( Y ; µ N ) is a free Z /N Z -module by Poincar´e duality. The ´etale cohomol-ogy groups H i ( Y ; µ N ) are the cohomology of a finite complex C • of finitefree Z /N Z -modules (see for example [7, p. 95, Theorem 4.9]). Thus, allbut possibly one of the Z /N Z -modules H i ( C • ) is free. It is then a standardfact in homological algebra that H i ( C • ) is a free Z /N Z -module for all i . Inparticular H ( Y ; µ N ) is free.(iii) Using (ii), H ( Y ; µ N ) is a free Z /N Z -module. By Poincar´e duality, itis enough to show that the class [ F ] of F in H ( Y ; µ N ) is primitive, i.e. isnot divisible by an integer ℓ > N , which we may assume prime.Arguing by contradiction, suppose that [ F ] = ℓγ for some prime ℓ and some γ ∈ H ( Y ; µ N ). Fix a point p ∈ B and let F be the corresponding fiber,which we assume is smooth. The image of [ F ] in H ( Y ; µ ℓ ) is zero, andhence, via the Kummer sequence, O Y ( F ) = L ⊗ ℓ for some line bundle L on34 . Then L restricts to an ℓ -torsion line bundle on every fiber of f and L has degree zero on every component of every fiber. If L | F is trivial on onesmooth fiber F , or equivalently on all smooth fibers, then by semicontinuity L | F has a section for every fiber F , smooth or not. Since there are nomultiple fibers, a standard argument (“Ramanujam’s lemma,” see e.g. [4,II.12.2] or [12, Exercise 1 p. 191]) shows that L | F = O F for all fibers F of f , and hence L = f ∗ λ for some line bundle λ on B . But this is clearlyimpossible, since then we would have f ∗ λ ⊗ ℓ = f ∗ O B ( p ). Applying f ∗ thengives λ ⊗ ℓ = O B ( p ) and hence ℓ · deg λ = 1, which is absurd.Thus the restriction of L to every smooth fiber has order ℓ . Let ν : Y ′ → Y be the µ ℓ cover defined by the ℓ th root L of O Y ( F ) and the natural sectionof O Y ( F ). By construction, ν ∗ O Y ′ = O Y ⊕ L − ⊕ · · · ⊕ L − ( ℓ − . Then Y ′ isa smooth surface and the induced morphism f ′ : Y ′ → B has generic fiberequal to a nontrivial ´etale cover of a genus one curve. Hence Y ′ is again agenus one fibration over B with Jacobian surface J ( Y ′ ) = X ′ . We are thenin the situation of the proof of (i), except that, as ( f ′ ) ∗ ( p ) = ν ∗ F = ℓG for some curve G on Y ′ isomorphic to F , Y ′ has the unique multiple fiber G over p of multiplicity ℓ , and G is tame. As before, if χ ( Y ; O Y ) = d , sothat the Euler number e ( Y ) = c ( Y ) = 12 d >
0, then e ( Y ′ ) = ℓe ( Y ) = 12 dℓ and hence χ ( Y ′ ; O Y ′ ) = dℓ .On the other hand, χ ( Y ′ ; O Y ′ ) = 1 − q ( Y ′ ) + p g ( Y ′ ). Since G is tame, R ( f ′ ) ∗ O Y ′ = L is a line bundle on B with deg L = deg( ω X ′ /B ) − < q ( Y ′ ) = h ( Y ′ ; O Y ′ ) = g ( B ) = q ( Y ). Since ν : Y ′ → Y is ´etaleaway from F , and is totally ramified at F , K Y ′ = ν ∗ K Y + ( ℓ − G . Since ν ∗ K Y | G is trivial and O Y ′ ( G ) | G is torsion of order ℓ , it is easy to check that p g ( Y ′ ) = h ( Y ′ ; K Y ′ ) = h ( Y ′ ; ν ∗ K Y ), and, again arguing as in the proofof (i), p g ( Y ′ ) = h ( Y ; K Y ) = p g ( Y ). But then χ ( Y ′ ; O Y ′ ) = 1 − q ( Y ′ ) + p g ( Y ′ ) = χ ( Y ; O Y ) = d , contradicting the fact that χ ( Y ′ ; O Y ′ ) = dℓ andthat d >
0. It follows that [ F ] is not divisible in H ( Y ; µ N ), completing theproof of (iii). Remark 4.7.
If in addition all fibers of f are irreducible, then, by theisomorphism R f ∗ µ n ∼ = R π ∗ µ n and an analysis of the Leray spectral se-quences for Y and for X , one can further show that H ( B ; R f ∗ µ n ) = 0.Using the Leray spectral sequence again, this gives an easier proof of (iii)under the additional assumption that all fibers of f are irreducible.Pontrjagin square gives a quadratic form ℘ : H ( X ; µ n ) → Z / n Z , whichinduces a similar quadratic form on H ( B ; R π ∗ µ n ) (also denoted by ℘ ).In both cases it can be computed by lifting classes modulo 2 n and squar-35ng using the cupproduct. It is straightforward to check that the isomor-phisms of Lemma 4.4 are compatible with Pontrjagin square. We then havethe following formula, special cases of which were established by Artin andSwinnerton-Dyer [2]: Lemma 4.8.
Suppose that f or equivalently π has a singular fiber. If α ∈ H ( B ; R π ∗ µ n ) and Y is the corresponding genus one fibration, with n -section D , then D ≡ − n d + ℘ ( α ) mod 2 n. Proof.
To compute ℘ ( α ), it suffices to lift α to a class in H ( B ; R π ∗ µ n ) andcompute its square. By Lemma 4.4, we can work with [ D ] ∈ H ( B ; R f ∗ µ n ).The class [ D ] ∈ H ( Y ; µ n ) does not induce a class in H ( B ; R f ∗ µ n )since its image in H ( B ; R f ∗ µ n ) is nonzero. To correct this, applyingProposition 4.6 with N = 2 n , there exists θ ∈ H ( Y ; µ n ) such that θ · F = 1.Thus the class [ D ] − nθ ∈ H ( Y ; µ n ) maps to 0 in H ( B ; R f ∗ µ n ) and soinduces a class in H ( B ; R f ∗ µ n ) which is clearly a lift of [ D ]. Computing,we find that ℘ ( α ) ≡ ( D − nσ ) = D − n ( D · θ ) + n θ mod 2 n ≡ D + n d mod 2 n, since by the Wu formula [23, Proposition 2.1], θ + K Y · θ ≡ D = − R f ∗ O Y ( − D ) − ( n + 2) d, we see that the n possible values of c ( R f ∗ O Y ( − D )) mod n , or equivalentlyof c ( f ∗ O Y ( D )) mod n , determine and are determined by D or by ℘ ( α ) mod2 n . Note that D + K Y · D = D + n ( d − ≡ D ≡ nd mod 2.This is consistent with the lemma above since − n d ≡ nd mod 2 and ℘ ( α ) ≡ A is a free Z /n Z -module, for example A = H ( P ; R π ∗ µ n ), then a class α ∈ A is primitive if the following holds: if n ′ is a positive integer dividing n and α = n ′ α ′ for some α ′ ∈ A , then n ′ = 1.Similarly, if Λ is a free Z -module, a class λ ∈ Λ is primitive if the followingholds: if λ = aλ ′ for some positive integer a and λ ′ ∈ Λ, then a = 1.36 heorem 4.9. Suppose that k = C . Let i ∈ Z / n Z satisfy i ≡ , andlet S d,n,i be the coarse moduli space of triples ( Y, f, D ) , where f : Y → P isa genus one fibration with Jacobian surface π : X → P , such that (i) χ ( Y ; O Y ) = χ ( X ; O X ) = d ; (ii) All fibers of f or equivalently π are irreducible; (iii) The class α ∈ H ( P ; R π ∗ µ n ) corresponding to Y is primitive and ℘ ( α ) = i .Then S d,n,i is irreducible.Proof. We may assume that d ≥
2. Let M d ⊆ M d be the coarse modulispace of elliptic surfaces π : X → P with χ ( X ; O X ) = d such that all fibersof π are irreducible. As we have seen in the discussion of Theorem 4.1, M d is irreducible. If t ∈ M d , we denote the corresponding elliptic surface by X t .Let S d,n be the coarse moduli space of triples ( Y, f, D ), where f : Y → P isa genus one fibration with Jacobian surface π : X → P satisfying (i) and (ii)in the statement of the theorem. There is a morphism S d,n → M d whosefiber over a general point t is H ( P ; R π ∗ µ n ) ∼ = [ F ] ⊥ / ( Z /n Z ) · [ F ], viewedas a subquotient of H ( X t ; µ n ). (In particular, if X t is general, then thepair ( X t , σ ) has no nontrivial automorphisms.) The irreducible componentsof S d,n then correspond to the orbits of the action of Γ = π ( M d , t ) on[ F ] ⊥ / ( Z /n Z ) · [ F ], which we can also identify with { σ, F } ⊥ . The actionof Γ preserves both the divisibility and the Pontrjagin square of a class α ∈ { σ, F } ⊥ , so it suffices to show that Γ acts transitively on the set ofprimitive classes α with ℘ ( α ) = i .In H ( X t ; Z ), the classes σ and F span a unimodular lattice, and { σ, F } ⊥ = Λ ∼ = (2 d − U ⊕ d ( − E ) , where U is the rank two hyperbolic lattice and − E is the negative of theroot lattice E . The sublattice { σ, F } ⊥ of H ( X t ; µ n ) is the mod n reductionof Λ, and if α is the reduction of an element λ ∈ Λ, then ℘ ( α ) = λ mod 2 n .The action of Γ on { σ, F } ⊥ ⊆ H ( X t ; µ n ) is the reduction of the action of Γon Λ. We begin by identifying this action, thanks to a result of L¨onne [17]based on work of Ebeling [8, 9]: Theorem 4.10.
The image of Γ in the orthogonal group O (Λ) contains theindex two subgroup O ∗ (Λ) of elements of real spinor norm one. roof. Begin with the degenerate Weierstrass equation y z = 4 x − g xz − g z defining a singular surface X ∈ P ( O P (2 d ) ⊕ O P (3 d ) ⊕ O P ), where g = 0 ∈ H ( P ; O P (4 d )) and g ∈ H ( P ; O P (6 d )) is a section vanishingto order 6 d − p ∈ P and hence vanishes simply at one otherpoint. In local coordinates, X has a singularity analytically of the form y = 4 x + z d − , which is of type E d − , in Arnold’s notation. Accordingto Table 3 in [8], the intersection pairing on the Milnor lattice of E d − isunimodular of rank 12 d − X is a generalregular elliptic surface with χ ( O X ) = d , one can identify the Milnor latticeof E d − with a sublattice of H ( X ; Z ), which is orthogonal to σ and F ,hence is contained in Λ and therefore equal to Λ since both are unimodular.The universal family of elliptic surfaces containing X fails to be a versaldeformation of the E d − singularity. In fact, the E d − singularity has a C ∗ -action, and it is easy to check directly that the universal family of ellipticsurfaces gives a versal deformation for the negative weight direction, andhence is transverse to the µ = constant stratum. Using results of Pinkham[21], one can also identify the Milnor fiber with X t − σ − F c , where X t isa smooth elliptic surface with χ ( O X t ) = d , σ is a section on X t , and F c isa cusp fiber. Thus the intersection pairing on the Milnor fiber is that of H ( X t − σ − F c ; Z ) and hence is isomorphic to Λ.Without appealing to the theory of deformations of singularities with C ∗ -action, one can argue directly (following [17]) as follows: By adding asmall term to g which does not vanish at p and a small linear term to g (inlocal coordinates), we obtain a new elliptic surface X s with local equation y = 4 x + a x + t d − + a t + s, where we view a and a as fixed and s as a parameter. The surface X s willhave a singular point ( ξ, η, τ ) ∈ X s exactly when when the partial derivatives12 x + a , 2 y , (6 d − t d − + a all vanish for ( x, y, t ) = ( ξ, η, τ ). If ξ is aroot of 12 x + a , τ is a root of (6 d − t d − + a and s = − (4 ξ + a ξ + τ d − + a τ ) = − a ξ − (cid:18) d − d − (cid:19) a τ, then the surface X s has an ordinary double point corresponding to ( x, y, t ) =( ξ, , τ ). One checks that, if a and a are general, then in this way we pro-duce 12 d − X s , each with a single ordinary double pointnear the original E d − singularity. For each such surface and ordinary dou-ble point, let δ be the corresponding vanishing cycle and let ∆ ⊆ Λ be theset of all such vanishing cycles. Then ∆ is the set of vanishing cycles for the38 d − singularity. As such, ∆ spans Λ and the Dynkin diagram correspond-ing to ∆ is connected. Hence, if Γ ∆ is the group generated by reflectionsabout the vanishing cycles in ∆, then ∆ is contained in a single Γ ∆ -orbit.Moreover, the Dynkin diagram for ∆ contains a certain subdiagram with6 vertices, which makes the pair (Λ , ∆) a complete vanishing lattice in thesense of [9]. It then follows by [9, Theorem 5.3.4] that Γ ∆ = O ∗ (Λ).On the other hand, it is easy to see that the deformations of X s areversal for the unique double point on X s . The monodromy associated tothis deformation acts on Λ as the reflection in the corresponding vanishingcycle. Hence the image of the monodromy group contains Γ ∆ and thus itcontains O ∗ (Λ). Remark 4.11.
Although the above proof analyzes the negative weight de-formations of the E d − singularity y = 4 x + z d − , it is in many waysmore natural to consider instead the singularity y = 4 x − c xz d − c z d ,whose Milnor fiber is diffeomorphic to X t − σ − F , where X t is a smoothelliptic surface with χ ( O X t ) = d , σ is a section, and F is a smooth fiber on X t isomorphic to the elliptic curve with equation y = 4 x − c x − c .Next we have the following standard result, due to Wall [24] in theunimodular case: Lemma 4.12.
Fix an even integer j . Then the group O ∗ (Λ) acts transitivelyon the set of primitive classes λ ∈ Λ with λ = j .Proof. By [24], the result holds with O (Λ) instead of O ∗ (Λ). In particular,fixing a hyperbolic summand U of Λ with standard basis ε, δ ( ε = δ = 0and ε · δ = 1), every primitive λ ∈ Λ with λ = j is equivalent under A ∈ O (Λ) to ε + ( j/ δ . If A / ∈ O ∗ (Λ), we can find a hyperbolic summand U ′ of Λ orthogonal to U and modify A by a reflection about an element of U ′ to adjust the spinor norm to be 1.To complete the proof of Theorem 4.9, it therefore suffices to prove: Lemma 4.13.
For every i ∈ Z / n Z with i ≡ , the group O ∗ (Λ) actstransitively on the set of primitive classes α ∈ Λ /n Λ with ℘ ( α ) = i .Proof. Fix an integer j whose reduction mod 2 n is i . By Lemma 4.12, itsuffices to show that, if α is a primitive class in Λ /n Λ with ℘ ( α ) = i , thenthere exists an integral lift ˜ α ∈ Λ such that ˜ α is primitive and ( ˜ α ) = j .First we claim that we can assume that ˜ α is primitive. Begin with anylift ˜ α of α to Λ. If ˜ α is not primitive, let ˜ α = aλ ′ , where a > ′ is primitive. Since α is primitive, a and n are relatively prime. We canassume by Lemma 4.12 that λ ′ is a primitive vector in a standard hyperbolicsummand U of Λ. Choosing a hyperbolic summand U ′ of Λ orthogonal to U and a primitive vector ε ′ ∈ U ′ with ( ε ′ ) = 0, the vector ˜ α + nε ′ is thena primitive vector in Λ lifting α .Let ( ˜ α ) = ℓ = j + 2 nk . Since ˜ α is primitive, we may as well assume,again by Lemma 4.12, that ˜ α = ( ℓ/ ε + δ , where ε, δ are a standard basisof a hyperbolic summand U of Λ. But then [( ℓ/ − nk ] ε + δ is another liftof α to Λ, and its square is j . This completes the proof of the lemma andhence of of Theorem 4.9. Throughout this section we consider only the case of elliptic fibrations withbase P . Definition 5.1.
A rank n vector bundle V on P is rigid if there exists aninteger a such that V ∼ = O P ( a ) k ⊕ O P ( a − n − k for some integer k >
0. Inother words, up to a twist by O P ( a ), V ∼ = O k P ⊕ O P ( − n − k with k > Lemma 5.2.
Let V be a rank n vector bundle on P with V = L ni =1 O P ( a i ) with a ≥ a ≥ · · · ≥ a n , or equivalently such that h ( P ; V ) = 0 but h ( P ; V ( − . Then V is rigid ⇐⇒ a n ≥ − ⇐⇒ h ( V ) = χ ( V ) ⇐⇒ h ( V ) = 0 . In terms of divisors on genus one fibrations, we have:
Lemma 5.3.
Let Y → P be a genus one fibration and let D be an f -nef divi-sor on Y such that D is effective but D − F is not effective. Then f ∗ O Y ( D ) is rigid ⇐⇒ π ∗ O X ( D ) = O k P ⊕ O P ( − n − k ⇐⇒ h ( Y ; O Y ( D )) = χ ( Y ; O Y ( D )) .Proof. Immediate from Lemma 5.2, the Leray spectral sequence, and thefact that R f ∗ O Y ( D ) = 0.Of course, f ∗ O Y ( D ) is rigid ⇐⇒ R f ∗ O Y ( − D ) is rigid. In this section,it will be slightly simpler to work with f ∗ O Y ( D ).The goal of this section is to show the following: Theorem 5.4.
Assume that k = C . Let d ≥ and let M d be the coarsemoduli space of elliptic surfaces π : X → P with χ ( O X ) = d and such thatall fibers of π are irreducible. Fix n ≥ . Then there exists an open dense ubspace of M d consisting of elliptic surfaces X such that, for every genusone fibration f : Y → P whose Jacobian surface is X and for every f -nefdivisor D on Y with D · F = n , the rank n vector bundle f ∗ O Y ( D ) is rigid. The proof will be by induction on d . In the case d = 2, we have: Lemma 5.5.
Let X be an elliptic K surface, let f : Y → P be a genusone fibration whose Jacobian surface is X , and let D be an f -nef divisor on Y with D · F = n . Then f ∗ O Y ( D ) is rigid.Proof. This follows easily from Remark 2.9. It is also easy to give a directproof in case D is minimal. In this case, there exists an irreducible curve C in | D | . From the exact sequence 0 → O Y ( − C ) → O Y → O C →
0, itfollows that h ( Y ; O Y ( − C )) = 0 and hence that h ( Y ; O Y ( C )) = 0. Thus h ( Y ; O Y ( D )) = χ ( Y ; O Y ( D )) and so f ∗ O Y ( D ) is rigid by Lemma 5.3.In the general case, the strategy is as follows. Using the discussion of theirreducible components from the previous section, it is enough to construct,for every integer k mod n , a single genus one fibration f : Y → P with χ ( O Y ) = d , with all fibers of f irreducible, and a divisor D on Y of relativedegree n such that f ∗ O Y ( D ) is rigid and c ( f ∗ O Y ( D )) ≡ k mod n . We willdo so by finding suitable degenerations of genus one fibrations Y to normalcrossings fibrations f : Y ′ ∪ X → P ∪ P , where Y ′ is a genus one fibrationwith χ ( O Y ′ ) = d − X is a rational elliptic surface, and Y ′ , X are gluedalong a fiber.To check rigidity via degenerations, we shall use the following lemma: Lemma 5.6.
Let φ : B → ∆ be a smooth surface, fibered over a smoothcurve ∆ , such that φ − ( t ) ∼ = P for t = t ∈ ∆ and such that φ − ( t ) hastwo components C and C , each isomorphic to P , glued at a point, so that φ − ( t ) has a single ordinary double point. Let V be a vector bundle on B such that V| C = O k P ⊕ O P ( − n − k , where k > , and V| C = O n P . Then,there exists a nonempty Zariski open subset U of ∆ such that, for all t ∈ U , V| φ − ( t ) is rigid, and in fact V| φ − ( t ) ∼ = O k P ⊕ O P ( − n − k .Proof. Let V t = V| φ − ( t ). By Lemma 5.2, it suffices to show that h ( V t ) = k and h ( V t ) = h ( V t ( − t ∈ U . First we claim that h ( V t ) = 0.This follows from the normalization exact sequence0 → V t → n ∗ ( V| C ⊕ V| C ) → ( C n ) p → , where p is the singular point of φ − ( t ) and n : C ∐ C → φ − ( t ) is thenormalization. Next, h ( V t ⊗ L ) = 0, where L is a line bundle on B which41estricts to O P ( −
1) on φ − ( t ), t ∈ U , restricts to O P ( −
1) on C , and istrivial on C , as again follows easily from the normalization exact sequence.So there is a nonempty Zariski open subset U of ∆ such that, for all t ∈ U , h ( V t ) = h ( V t ( − χ ( V t ) = χ ( V t ) = k >
0, and thus h ( V t ) = k as well.It is then enough to establish the following: Claim 5.7.
Fix an integer k , < k ≤ n and an integer d ≥ . Then thereis a family Y → B → ∆ where Y is a smooth threefold and B is a smoothsurface, ∆ is a smooth curve as in Lemma 5.6, and a point t ∈ ∆ with thefollowing properties. Denote by B t the fiber of B over t ∈ ∆ , and similarlyfor Y t , and let f t : Y t → B t be the induced morphism. Then: (i) B t ∼ = P for t = t and B t ∼ = P ∪ P meeting normally; (ii) The induced morphism f t : Y t → B t gives Y t the structure of a genusone fibration with all fibers irreducible and χ ( O Y t ) = d , for t = t ; (iii) Y t = Y ′ ∪ X , where Y ′ is a genus one fibration with all fibers irreducibleand χ ( O Y ′ ) = d − , X is a rational elliptic surface with all fibersirreducible, Y ′ and X meet normally along a smooth fiber for bothof the given fibrations, and the morphism f t = f ′ ∪ g : Y ′ ∪ X → P ∪ P induces the two given fibrations on the two components of thenormalization of Y t ; (iv) There exists a Cartier divisor D on Y , such that D| Y ′ = D ′ sat-isfies: ( f ′ ) ∗ O Y ′ ( D ′ ) = O k P ⊕ O P ( − n − k and D| X = D satisfies: g ∗ O X ( D ) = O n P . We prove the claim in several steps. First, we can find the genus onefibration f ′ : Y ′ → P and the divisor D ′ by induction on d , beginning withthe case d = 2. Next we find appropriate divisors on rational elliptic surfaces: Theorem 5.8.
Let π : X → P be a rational elliptic surface and let n ∈ Z , n ≥ . Then there exists a divisor D on X such that D · F = n and π ∗ O X ( D ) = O n P . The proof follows from the next two lemmas:
Lemma 5.9.
Let π : X → P be a rational elliptic surface and let n ∈ Z , n ≥ . Let D be a smooth rational curve in X such that D = n − . Then π ∗ O X ( D ) = O n P . roof. It follows from the exact sequence0 → O X → O X ( D ) → O P ( n − → h ( O X ( D )) = 0, and clearly h ( O X ( D )) = h ( O X ( − D − F )) = 0.From the exact sequence0 → O X ( − D ) → O X → O D → , it follows that h ( O X ( − D )) = 0, and hence that h ( O X ( D + K X )) = h ( O X ( D − F )) = 0.By adjunction, − D + D · K X = n − D · K X = n − − D · F . Thus D · F = n . It follows that χ ( O X ( D )) = h ( O X ( D )) = ( D − D · K X ) + 1 = (2 n −
2) + 1 = n .By Lemma 5.3, π ∗ O X ( D ) = O k P ⊕ O P ( − n − k is rigid, and k = h ( P ; π ∗ O X ( D )) = h ( X ; O X ( D )) = n. Thus π ∗ O X ( D ) = O n P . Lemma 5.10.
Let π : X → P be a rational elliptic surface and let n ∈ Z , n ≥ . Then there exists a smooth rational curve D in X such that D = n − .Proof. If n = 1, then we can just take a section of X , or equivalently anexceptional curve. Otherwise, there is a blowdown ρ : X → ¯ X , where ¯ X is a smooth minimal rational surface with a smooth anticanonical divisor,and hence ¯ X = F , F , or P . Thus X dominates F , F , or F . Moreover, X is a blowup of either F , F , or F at 8 points, and if X is a blowup of F , then the first point of the blowup does not lie on the negative section.It then follows easily that X either simultaneously dominates F and F ,or X simultaneously dominates F and F . It thus suffices to show: if n iseven and n ≥
2, then there exist base point free linear systems on F and F whose general members are smooth rational curves D with D = n − n is odd and n ≥
3, then there exists a linear system on F whosegeneral member is a smooth rational curve D with D = n −
2. The propertransform in X of a general D will then have the desired properties.If n = 2, then we can take D to be a general fiber of the ruling on F a , a = 0 , ,
2. So we can assume n ≥
3. In case n = 2 a is even, a ≥
2, and X dominates F , choose the base point free linear system | σ + aF | , where σ is the negative section. Then the general member D of | σ + aF | is smoothand satisfies D = 2 a − n −
2. The case where X dominates F is43imilar, using the linear system | σ + ( a − F | , where σ , F are the classesof the two rulings on F . If n = 2 a + 1 is odd and n ≥
3, so that a ≥ | σ + aF | on F , where σ is thenegative section. The general member D of | σ + aF | is smooth and satisfies D = 2 a − n − Y of genus one fibrations: Theorem 5.11.
Let π : X → P be a smooth rational elliptic surface, let π ′ : X ′ → P be an elliptic surface with χ ( O X ′ ) = d − , let F ⊆ X and F ′ ⊆ X ′ be two smooth fibers which are isomorphic as elliptic curves. Thenthere exists a family X → B → ∆ where X is a smooth threefold and B is asmooth surface, ∆ is a smooth curve and t ∈ ∆ as in Lemma 5.6, with thefollowing properties: let B t be the fiber of B over t ∈ ∆ , and similarly for X t , and let π t : X t → B t be the induced morphism. Then: (i) B t ∼ = P for t = t and B t ∼ = P ∪ P meeting normally; (ii) The induced morphism π t : X t → B t ∼ = P realizes X t as an ellipticsurface over P with χ ( O X t ) = d , for t = t ; (iii) X t = X ′ ∪ X , where X ′ and X meet normally along the curves F ∼ = F ′ ,and the morphism π t = π ′ ∪ π : X ′ ∪ X → P ∪ P induces the twogiven fibrations on the two components of the normalization of X t .Proof. The surface X ′ is defined by sections g ′ ∈ H ( P ; O P (4( d − g ′ ∈ H ( P ; O P (6( d − p ∈ P be the point corresponding tothe fiber F ′ . Since O P (4 d ) ⊗ O P ( − p ) ∼ = O P (4( d − O P (6 d ) ⊗O P ( − p ) ∼ = O P (6( d − g ′ defines a section g ∈ H ( P ; O P (4 d )) van-ishing to order 4 at p , and likewise g ′ defines a section g ∈ H ( P ; O P (6 d ))vanishing to order 6 at p . The Weierstrass equation y z = 4 x − g xz − g z defines a hypersurface X in the P -bundle P ( O P (2 d ) ⊕ O P (3 d ) ⊕ O P ), witha cuspidal fiber E over p , and X has a simple elliptic singularity at the cuspof E . More precisely, fixing a local coordinate u on P at p and a local sectionof O P (1) at p , and hence of O P (4 d ) and O P (6 d ), the singularity definedby the above Weierstrass equation is analytically y = 4 x − c xu − c u .For simplicity, we shall just consider the case where c = 0, i.e. where the j -invariant of the fiber F is not 1728.Because the restriction homomorphism H ( P ; O P (4 d )) → O P (4 d ) /u O P (4 d )44s surjective, for i = 0 , , , h ( i )2 ∈ H ( P ; O P (4 d )) suchthat, using the local coordinate u at p and the fixed trivialization of O P (4 d ))chosen above, h ( i )2 ( u ) = u i + O ( u ). Likewise, for i = 0 , . . . ,
5, there existsections h ( i )3 ∈ H ( P ; O P (6 d )) such that, using the local coordinate u at p and the fixed trivialization of O P (6 d )) chosen above, h ( i )3 ( u ) = u i + O ( u ).Let v be a coordinate of A . For fixed α , . . . , α , β , . . . , β ∈ k , consider thefamily X ⊆ P ( O P (2 d ) ⊕O P (3 d ) ⊕O P ) × A of elliptic surfaces parametrizedby v ∈ A defined by y z = 4 x − G ( v ) xz − G ( v ) z , where G ( v ) = g + X i =1 α i v i h (4 − i )2 ; G ( v ) = g + X i =1 β i v i h (6 − i )3 . Note that there is an induced morphism
X → P × A as well as a morphism X → A whose fiber over 0 is X . It is easy to check that, if the α , . . . , β are not all 0, then the general fiber is an elliptic surface over P with atworst rational double points. There is a point q ∈ X lying over ( p,
0) andcorresponding to x = y = 0 , z = 1; the local equation for X at q is y = 4 x − ( c u + X i =1 α i v i u − i + O ( u )) x − ( c u + X i =1 β i v i u − i + O ( u )) . Make a weighted blowup of the open subset { z = 0 } of P ( O P (2 d ) ⊕O P (3 d ) ⊕O P ) × A with coordinates u, v, x, y , where u and v have weight 1, x hasweight 2 and y has weight 3. Thus the exceptional divisor is a weightedprojective space P (1 , , , e X of X in this weighted blowup has the effect of resolving thesimple elliptic singularity at q ∈ X , and in fact is the minimal resolution ofthe simple elliptic singularity, as well as resolving the cusp singularity on E .If ˜ X is the proper transform of X , then the new exceptional divisor on ˜ X isan elliptic curve ˜ F of self-intersection −
1, the proper transform of E is anexceptional curve E , and contracting E gives a morphism ˜ X → X ′ which isthe blowup of X ′ at a point on the fiber F .The exceptional divisor of the morphism e X → X is a hypersurface ˆ X in the weighted projective space P (1 , , ,
3) with homogeneous coordinates u, v, x, y . As ˆ X is defined by the homogeneous degree 6 equation y = 4 x − ( c u + X i =1 α i v i u − i ) x − ( c u + X i =1 β i v i u − i ) ,
45t is a degree one (generalized) del Pezzo surface in P (1 , , , α i , β j , one can arrangethat ˆ X is an arbitrary del Pezzo surface subject to the condition that thecurve defined by u = 0 is isomorphic to F . If ˆ X is smooth (no rational doublepoints), then X will be smooth as well, at least in a neighborhood of thefiber over 0. By a standard construction in threefold birational geometry(a “Type I modification,” see for example [14, p. 13]), one can flip theexceptional curve E on ˜ X to ˆ X . The new fiber over 0 ∈ A then consistsof X ′ together with the blowup X of the del Pezzo surface ˆ X at the basepoint of | − K ˆ X | ( x = 1 , y = 2 , u = v = 0), which is then a rational ellipticsurface. The construction can be summarized in the following picture:˜ X ∪ ˆ X ⊆ e X X ⊇ X ′ ∪ X ↓ ↓ ↓ X ⊆ X B↓ ւ P × A The birational morphism e X X is the Type I modification, which con-tracts the exceptional curve E on ˜ X and blows up the corresponding pointof ˆ X to obtain a rational elliptic surface X . The inverse image of ( p, ∈ P × A in X is the cuspidal fiber E . Thus the preimage of ( p,
0) in theweighted blowup of X consists of the union of the exceptional divisor ˆ X and the proper transform of E , namely E . After flipping E , the inverseimage of ( p, ∈ P × A is just the divisor X , and hence the morphism X → P × A factors through the blowup B of P × A at the point ( p, ∈ A of the morphism X → A is X ′ ∪ X , and for t = 0, t in a nonempty Zariski open subset of A , the fiber over t of themorphism X → A is a smooth elliptic surface. It is easy to check that theinduced morphism X ′ ∪ X → P ∪ P induces the given fibrations on X ′ and X . Replace A by the nonempty open set ∆ which is the complementof points other than 0 where X → A fails to be smooth, and X , B by therespective preimages of ∆. It is then straightforward to see that X → B isas claimed.We now complete the proof of Claim 5.7 and thus of Theorem 5.4. Let d ≥
3. By induction there exists an elliptic surface π ′ : X ′ → P with χ ( O X ′ ) = d − ≥
2, with all fibers of π ′ irreducible, and such that thereexists a class ξ ′ ∈ H ( X ′ ; R ( π ′ ) ∗ µ n ) so that the corresponding pair ( Y ′ , D ′ )satisfies: ( f ′ ) ∗ O Y ′ ( D ′ ) = O k P ⊕ O P ( − n − k . Let π : X → P be a ratio-nal elliptic surface with all fibers irreducible, let D be a divisor of fiber46egree n on X such that π ∗ O X ( D ) = O n P (whose existence is guaranteedby Theorem 5.8), and let ξ ∈ H ( X ; R π ∗ µ n ) be the class corresponding tothe pair ( X, D ). Using the section σ ⊆ X , we can identify H ( X ; R π ∗ µ n )with { σ, F } ⊥ ⊆ H ( X ; µ n ), and similarly for X ′ . Let X → B → ∆ be thefamily constructed in Theorem 5.11. From the Mayer-Vietoris sequence for X = X ′ ∪ X , namely0 → ( µ n ) X → ( µ n ) X ′ ⊕ ( µ n ) X → ( µ n ) F → , the pair ( ξ ′ , ξ ) induces an element in H ( X ; µ n ) ⊕ H ( X ′ ; µ n ) which isorthogonal to the classes of F and F ′ , and hence an element of λ ∈ H ( X ; µ n ) orthogonal to the class of a fiber in each surface.By the proper base change theorem, after replacing ∆ by an ´etale cover,which we continue to denote by ∆, there exists a class λ ∈ H ( X ; µ n ) whoserestriction to H ( X ; µ n ) is λ , and hence λ · [ F ] = 0 for every fiber F of themorphism X → B . So finally there is an induced element of H ( X ; R π ∗ µ n ),also denoted by λ , which restricts in the appropriate sense on X ′ and X togive the classes ξ ′ and ξ , respectively.By the remarks at the beginning of the last section, since all fibers of themorphism X → B are irreducible, the class λ ∈ H ( X ; R π ∗ µ n ) correspondsto a principal homogeneous space Y → B , together with a divisor D on Y ofrelative degree n . By construction, it is clear that the conditions of Claim 5.7are satisfied. This completes the proof of Claim 5.7 and Theorem 5.4. References [1] A. Altman and S. Kleiman,
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On the orthogonal groups of unimodular quadratic forms ,Math. Ann. (1962), 328–338.Department of MathematicsColumbia UniversityNew York, NY 10027USA [email protected], [email protected]@math.columbia.edu, [email protected]